Scalable neutral atom based quantum computing

ABSTRACT

The present disclosure provides methods and systems for performing non-classical computations. The methods and systems generally use a plurality of spatially distinct optical trapping sites to trap a plurality of atoms, one or more electromagnetic delivery units to apply electromagnetic energy to one or more atoms of the plurality to induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state, one or more entanglement units to quantum mechanically entangle at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality, and one or more readout optical units to perform measurements of the superposition states to obtain the non-classical computation.

CROSS-REFERENCE

This application is a continuation of U.S. application Ser. No.16/405,877, filed May 7, 2019, which claims the benefit of U.S.Provisional Patent Application No. 62/760,781, filed on Nov. 13, 2018,and U.S. Provisional Patent Application No. 62/815,985, filed on Mar. 8,2019, each of which is entirely incorporated herein by reference for allpurposes.

STATEMENT AS TO FEDERALLY SPONSORED RESEARCH

This invention was made with the support of the United States Governmentunder Small Business Innovation Research Grant No. 1843926 awarded bythe National Science Foundation. The United States Government hascertain rights in this invention.

BACKGROUND

Quantum computers typically make use of quantum-mechanical phenomena,such as superposition and entanglement, to perform operations on data.Quantum computers may be different from digital electronic computersbased on transistors. For instance, whereas digital computers requiredata to be encoded into binary digits (bits), each of which is always inone of two definite states (0 or 1), quantum computation uses quantumbits (qubits), which can be in superpositions of states.

SUMMARY

Recognized herein is the need for methods and systems for performingnon-classical computations.

The present disclosure provides systems and methods for utilizing atoms(such as neutral or uncharged atoms) to perform non-classical or quantumcomputations. The atoms may be optically trapped in large arrays.Quantum mechanical states of the atoms (such as hyperfine states ornuclear spin states of the atoms) may be configured to function asquantum bit (qubit) basis states. The qubit states may be manipulatedthrough interaction with optical, radiofrequency, or otherelectromagnetic radiation, thereby performing the non-classical orquantum computations.

In an aspect, the present disclosure provides a system for performing anon-classical computation, comprising: one or more optical trappingunits configured to generate a plurality of spatially distinct opticaltrapping sites, the plurality of optical trapping sites configured totrap a plurality of atoms, wherein the plurality of atoms are qubits,and wherein the plurality of atoms comprise greater than 60 atoms; oneor more electromagnetic delivery units configured to applyelectromagnetic energy to one or more atoms of the plurality of atoms,to induce the one or more atoms to adopt one or more superpositionstates of a first atomic state and at least a second atomic state thatis different from the first atomic state; one or more entanglement unitsconfigured to quantum mechanically entangle at least a subset of the oneor more atoms in the one or more superposition states with at leastanother atom of the plurality of atoms; and one or more readout opticalunits configured to perform one or more measurements of the one or moresuperposition states to obtain at least a portion of the non-classicalcomputation. The first atomic state may comprise a first hyperfineelectronic state and the second atomic state may comprise a secondhyperfine electronic state that is different from the first hyperfineelectronic state. The first atomic state may comprise a first nuclearspin state and the second atomic state may comprise a second nuclearspin state that is different from the first nuclear spin state. Theplurality of atoms may comprise neutral atoms. The plurality of atomsmay comprise a Group II element. The first and second atomic states maycomprise first and second nuclear spin states of a nucleus comprising anuclear spin greater than ½. The subset of the one or more atoms in theone or more superposition states and the another atom may be quantummechanically entangled with a coherence lifetime of at least 1 second.The plurality of atoms may comprise a temperature of at most 10microkelvin (μK). Each optical trapping site of the plurality of opticaltrapping sites may be spatially separated from each other opticaltrapping site by at least 200 nanometers (nm). The one or more opticaltrapping units may comprise one or more spatial light modulators (SLMs)configured to generate the plurality of optical trapping sites.

The one or more SLMs may comprise a digital micromirror device (DMDs) ora liquid crystal on silicon (LCoS) device. The one or more opticaltrapping units may comprise one or more light sources configured to emitlight tuned to one or more magic wavelengths corresponding to theplurality of atoms. The one or more optical trapping units may compriseone or more imaging units configured to obtain one or more images of aspatial configuration of the plurality of atoms trapped within theoptical trapping sites. The system may further comprise one or morespatial configuration artificial intelligence (AI) units configured toperform one or more AI operations to determine the spatial configurationof the plurality of atoms trapped within the optical trapping sitesbased on the one or more images. The one or more optical trapping unitsmay comprise one or more atom rearrangement units configured to impartan altered spatial arrangement of the plurality of atoms trapped withthe optical trapping sites based on he one or more images. The systemmay further comprise one or more spatial arrangement artificialintelligence (AI) units configured to perform one or more AI operationsto determine the altered spatial arrangement of the plurality of atomstrapped within the optical trapping sites based on the one or moreimages. The system may further comprise one or more said statepreparation units configured to cool the plurality of atoms to a firstdistribution of atomic states. The system may further comprise one ormore optical pumping units configured to emit light to optically pumpone or more atoms of the plurality of atoms from the first distributionof atomic states to a pure or nearly pure atomic state. The system mayfurther comprise one or more coherent driving units configured tocoherently drive the one or more atoms from the non-equilibrium atomicstate or the non-equilibrium distribution of atomic states to the firstor second atomic state. The one or more electromagnetic delivery unitsmay comprise one or more members selected from the group consisting of aspatial light modulator (SLM), an acousto-optic device (AOD), and anacousto-optic modulator (AOM), which one or more members are configuredto selectively apply the electromagnetic energy to one or more atoms ofthe plurality of atoms. The SLM may comprise a digital micromirrordevice (DMD) or a liquid crystal on silicon (LCoS) device. The one ormore electromagnetic delivery units may be configured to implement oneor more single-qubit gate operations on the one or more qubits. Thesystem may further comprise one or more atom reservoirs configured tosupply one or more replacement atoms to replace one or more atoms at oneor more optical trapping sites of the plurality of optical trappingsites upon loss of the one or more atoms from the one or more opticaltrapping sites. The system may further comprise one or more atommovement units configured to move the one or more replacement atoms tothe one or more optical trapping sites. The subset of the one or moreatoms in the one or more superposition states and the another atom maybe quantum mechanically entangled through a magnetic dipole interaction,an electric dipole interaction, or an induced electric dipoleinteraction. The one or more entanglement units may comprise one or moreRydberg excitation units configured to electronically excite the subsetof one or more atoms in the one or more superposition states to aRydberg state, thereby forming one or more Rydberg atoms, or tocontinuously drive the one or more atoms in the one or moresuperposition states via a transition to a Rydberg state, therebyforming one or more dressed Rydberg atoms. The one or more Rydbergexcitation units may be configured to induce one or more quantummechanical entanglements between the one or more Rydberg atoms ordressed Rydberg atoms and the another atom, the another atom located ata distance of no more than 10 micrometers (μm) from the one or moreRydberg atoms or dressed Rydberg atoms. The one or more Rydberg unitsmay be configured to drive the one or more Rydberg atoms or dressedRydberg atoms to a lower-energy atomic state, thereby forming one ormore two-qubit units. The one or more electromagnetic delivery units maybe configured to implement one or more two-qubit gate operations on theone or more two-qubit units. The system may be operatively coupled to adigital computer over a cloud computing network.

Another aspect of the present disclosure provides a non-transitorycomputer readable medium comprising machine executable code that, uponexecution by one or more computer processors, implements any of themethods above or elsewhere herein.

Another aspect of the present disclosure provides a system comprisingone or more computer processors and computer memory coupled thereto. Thecomputer memory comprises machine executable code that, upon executionby the one or more computer processors, implements any of the methodsabove or elsewhere herein.

Additional aspects and advantages of the present disclosure will becomereadily apparent to those skilled in this art from the followingdetailed description, wherein only illustrative embodiments of thepresent disclosure are shown and described. As will be realized, thepresent disclosure is capable of other and different embodiments, andits several details are capable of modifications in various obviousrespects, all without departing from the disclosure. Accordingly, thedrawings and description are to be regarded as illustrative in nature,and not as restrictive.

INCORPORATION BY REFERENCE

All publications, patents, and patent applications mentioned in thisspecification are herein incorporated by reference to the same extent asif each individual publication, patent, or patent application wasspecifically and individually indicated to be incorporated by reference.To the extent publications and patents or patent applicationsincorporated by reference contradict the disclosure contained in thespecification, the specification is intended to supersede and/or takeprecedence over any such contradictory material.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth with particularity inthe appended claims. A better understanding of the features andadvantages of the present invention will be obtained by reference to thefollowing detailed description that sets forth illustrative embodiments,in which the principles of the invention are utilized, and theaccompanying drawings (also “Figure” and “FIG.” herein), of which:

FIG. 1 shows a computer control system that is programmed or otherwiseconfigured to implement methods provided herein.

FIG. 2 shows an example of a system for performing a non-classicalcomputation.

FIG. 3A shows an example of an optical trapping unit.

FIG. 3B shows an example of a plurality of optical trapping sites.

FIG. 3C shows an example of an optical trapping unit that is partiallyfilled with atoms.

FIG. 3D shows an example of an optical trapping unit that is completelyfilled with atoms.

FIG. 4 shows an example of an electromagnetic delivery unit.

FIG. 5 shows an example of a state preparation unit.

FIG. 6 shows a flowchart for an example of a first method for performinga non-classical computation.

FIG. 7 shows a flowchart for an example of a second method forperforming a non-classical computation.

FIG. 8 shows a flowchart for an example of a third method for performinga non-classical computation.

FIG. 9 shows an example of a qubit comprising a 3P2 state ofstrontium-87.

DETAILED DESCRIPTION

While various embodiments of the invention have been shown and describedherein, it will be obvious to those skilled in the art that suchembodiments are provided by way of example only. Numerous variations,changes, and substitutions may occur to those skilled in the art withoutdeparting from the invention. It should be understood that variousalternatives to the embodiments of the invention described herein may beemployed.

Unless otherwise defined, all technical terms used herein have the samemeaning as commonly understood by one of ordinary skill in the art towhich this invention belongs. As used in this specification and theappended claims, the singular forms “a,” “an,” and “the” include pluralreferences unless the context clearly dictates otherwise. Any referenceto “or” herein is intended to encompass “and/or” unless otherwisestated.

Whenever the term “at least,” “greater than,” or “greater than or equalto” precedes the first numerical value in a series of two or morenumerical values, the term “at least,” “greater than” or “greater thanor equal to” applies to each of the numerical values in that series ofnumerical values. For example, greater than or equal to 1, 2, or 3 isequivalent to greater than or equal to 1, greater than or equal to 2, orgreater than or equal to 3.

Whenever the term “no more than,” “less than,” “less than or equal to,”or “at most” precedes the first numerical value in a series of two ormore numerical values, the term “no more than,” “less than,” “less thanor equal to,” or “at most” applies to each of the numerical values inthat series of numerical values. For example, less than or equal to 3,2, or 1 is equivalent to less than or equal to 3, less than or equal to2, or less than or equal to 1.

Where values are described as ranges, it will be understood that suchdisclosure includes the disclosure of all possible sub-ranges withinsuch ranges, as well as specific numerical values that fall within suchranges irrespective of whether a specific numerical value or specificsub-range is expressly stated.

As used herein, the terms “artificial intelligence,” “artificialintelligence procedure”, “artificial intelligence operation,” and“artificial intelligence algorithm” generally refer to any system orcomputational procedure that may take one or more actions to enhance ormaximize a chance of achieving a goal. The term “artificialintelligence” may include “generative modeling,” “machine learning”(ML), and/or “reinforcement learning” (RL).

As used herein, the terms “machine learning,” “machine learningprocedure,” “machine learning operation,” and “machine learningalgorithm” generally refer to any system or analytical and/orstatistical procedure that may progressively improve computerperformance of a task. Machine learning may include a machine learningalgorithm. The machine learning algorithm may be a trained algorithm.Machine learning (ML) may comprise one or more supervised,semi-supervised, or unsupervised machine learning techniques. Forexample, an ML algorithm may be a trained algorithm that is trainedthrough supervised learning (e.g., various parameters are determined asweights or scaling factors). ML may comprise one or more of regressionanalysis, regularization, classification, dimensionality reduction,ensemble learning, meta learning, association rule learning, clusteranalysis, anomaly detection, deep learning, or ultra-deep learning. MLmay comprise, but is not limited to: k-means, k-means clustering,k-nearest neighbors, learning vector quantization, linear regression,non-linear regression, least squares regression, partial least squaresregression, logistic regression, stepwise regression, multivariateadaptive regression splines, ridge regression, principle componentregression, least absolute shrinkage and selection operation, leastangle regression, canonical correlation analysis, factor analysis,independent component analysis, linear discriminant analysis,multidimensional scaling, non-negative matrix factorization, principalcomponents analysis, principal coordinates analysis, projection pursuit,Sammon mapping, t-distributed stochastic neighbor embedding,AdaBoosting, boosting, gradient boosting, bootstrap aggregation,ensemble averaging, decision trees, conditional decision trees, boosteddecision trees, gradient boosted decision trees, random forests, stackedgeneralization, Bayesian networks, Bayesian belief networks, naïveBayes, Gaussian naïve Bayes, multinomial naïve Bayes, hidden Markovmodels, hierarchical hidden Markov models, support vector machines,encoders, decoders, auto-encoders, stacked auto-encoders, perceptrons,multi-layer perceptrons, artificial neural networks, feedforward neuralnetworks, convolutional neural networks, recurrent neural networks, longshort-term memory, deep belief networks, deep Boltzmann machines, deepconvolutional neural networks, deep recurrent neural networks, orgenerative adversarial networks.

As used herein, the terms “reinforcement learning,” “reinforcementlearning procedure,” “reinforcement learning operation,” and“reinforcement learning algorithm” generally refer to any system orcomputational procedure that may take one or more actions to enhance ormaximize some notion of a cumulative reward to its interaction with anenvironment. The agent performing the reinforcement learning (RL)procedure may receive positive or negative reinforcements, called an“instantaneous reward”, from taking one or more actions in theenvironment and therefore placing itself and the environment in variousnew states.

A goal of the agent may be to enhance or maximize some notion ofcumulative reward. For instance, the goal of the agent may be to enhanceor maximize a “discounted reward function” or an “average rewardfunction”. A “Q-function” may represent the maximum cumulative rewardobtainable from a state and an action taken at that state. A “valuefunction” and a “generalized advantage estimator” may represent themaximum cumulative reward obtainable from a state given an optimal orbest choice of actions. RL may utilize any one of more of such notionsof cumulative reward. As used herein, any such function may be referredto as a “cumulative reward function”. Therefore, computing a best oroptimal cumulative reward function may be equivalent to finding a bestor optimal policy for the agent.

The agent and its interaction with the environment may be formulated asone or more Markov Decision Processes (MDPs), for example. The RLprocedure may not assume knowledge of an exact mathematical model of theMDPs. The MDPs may be completely unknown, partially known, or completelyknown to the agent. The RL procedure may sit in a spectrum between thetwo extents of “model-based” or “model-free” with respect to priorknowledge of the MDPs. As such, the RL procedure may target large MDPswhere exact methods may be infeasible or unavailable due to an unknownor stochastic nature of the MDPs.

The RL procedure may be implemented using one or more computerprocessors described herein. The digital processing unit may utilize anagent that trains, stores, and later on deploys a “policy” to enhance ormaximize the cumulative reward. The policy may be sought (for instance,searched for) for a period of time that is as long as possible ordesired. Such an optimization problem may be solved by storing anapproximation of an optimal policy, by storing an approximation of thecumulative reward function, or both. In some cases, RL procedures maystore one or more tables of approximate values for such functions. Inother cases, RL procedure may utilize one or more “functionapproximators”.

Examples of function approximators may include neural networks (such asdeep neural networks) and probabilistic graphical models (e.g. Boltzmannmachines, Helmholtz machines, and Hopfield networks). A functionapproximator may create a parameterization of an approximation of thecumulative reward function. Optimization of the function approximatorwith respect to its parameterization may consist of perturbing theparameters in a direction that enhances or maximizes the cumulativerewards and therefore enhances or optimizes the policy (such as in apolicy gradient method), or by perturbing the function approximator toget closer to satisfy Bellman's optimality criteria (such as in atemporal difference method).

During training, the agent may take actions in the environment to obtainmore information about the environment and about good or best choices ofpolicies for survival or better utility. The actions of the agent may berandomly generated (for instance, especially in early stages oftraining) or may be prescribed by another machine learning paradigm(such as supervised learning, imitation learning, or any other machinelearning procedure described herein). The actions of the agent may berefined by selecting actions closer to the agent's perception of what anenhanced or optimal policy is. Various training strategies may sit in aspectrum between the two extents of off-policy and on-policy methodswith respect to choices between exploration and exploitation.

As used herein, the terms “non-classical computation,” “non-classicalprocedure,” “non-classical operation,” any “non-classical computer”generally refer to any method or system for performing computationalprocedures outside of the paradigm of classical computing. Anon-classical computation, non-classical procedure, non-classicaloperation, or non-classical computer may comprise a quantum computation,quantum procedure, quantum operation, or quantum computer.

As used herein, the terms “quantum computation,” “quantum procedure,”“quantum operation,” and “quantum computer” generally refer to anymethod or system for performing computations using quantum mechanicaloperations (such as unitary transformations or completely positivetrace-preserving (CPTP) maps on quantum channels) on a Hilbert spacerepresented by a quantum device. As such, quantum and classical (ordigital) computation may be similar in the following aspect: bothcomputations may comprise sequences of instructions performed on inputinformation to then provide an output. Various paradigms of quantumcomputation may break the quantum operations down into sequences ofbasic quantum operations that affect a subset of qubits of the quantumdevice simultaneously. The quantum operations may be selected based on,for instance, their locality or their ease of physical implementation. Aquantum procedure or computation may then consist of a sequence of suchinstructions that in various applications may represent differentquantum evolutions on the quantum device. For example, procedures tocompute or simulate quantum chemistry may represent the quantum statesand the annihilation and creation operators of electron spin-orbitals byusing qubits (such as two-level quantum systems) and a universal quantumgate set (such as the Hadamard, controlled-not (CNOT), and π/8 rotation)through the so-called Jordan-Wigner transformation or Bravyi-Kitaevtransformation.

Additional examples of quantum procedures or computations may includeprocedures for optimization such as quantum approximate optimizationalgorithm (QAOA) or quantum minimum finding. QAOA may compriseperforming rotations of single qubits and entangling gates of multiplequbits. In quantum adiabatic computation, the instructions may carrystochastic or non-stochastic paths of evolution of an initial quantumsystem to a final one.

Quantum-inspired procedures may include simulated annealing, paralleltempering, master equation solver, Monte Carlo procedures and the like.Quantum-classical or hybrid algorithms or procedures may comprise suchprocedures as variational quantum eigensolver (VQE) and the variationaland adiabatically navigated quantum eigensolver (VanQver).

A quantum computer may comprise one or more adiabatic quantum computers,quantum gate arrays, one-way quantum computers, topological quantumcomputers, quantum Turing machines, quantum annealers, Ising solvers, orgate models of quantum computing.

Systems for Performing a Non-Classical Computation

In an aspect, the present disclosure provides a system for performing anon-classical computation. The system may comprise one or more opticaltrapping units configured to generate a plurality of spatially distinctoptical trapping sites. The plurality of optical trapping sites may beconfigured to trap a plurality of atoms. The plurality of atoms maycomprise greater than 60 atoms. The system may also comprise one or moreelectromagnetic delivery units configured to apply electromagneticenergy to one or more atoms of the plurality of atoms, thereby inducingthe one or more atoms to adopt one or more superposition states of afirst atomic state and at least a second atomic state that is differentfrom the first atomic state

The system may also include one or more entanglement units configured toquantum mechanically entangle at least a subset of the one or more atomsin the one or more superposition states with at least another atom ofthe plurality of atoms, and one or more readout optical units configuredto perform one or more measurements of the one or more superpositionstate to obtain the non-classical computation.

Reference will now be made to the figures, wherein like numerals referto like parts throughout. It will be appreciated that the figures arenot necessarily drawn to scale.

FIG. 2 shows an example of a system 200 for performing a non-classicalcomputation. The non-classical computation may comprise a quantumcomputation. The quantum computation may comprise a gate-model quantumcomputation.

The system 200 may comprise one or more optical trapping units 210. Theoptical trapping units may comprise any optical trapping unit describedherein, such as an optical trapping unit described herein with respectto FIG. 3A. The optical trapping units may be configured to generate aplurality of spatially distinct optical trapping sites. For instance,the optical trapping units may be configured to generate at least about10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700,800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000,10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000,100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000,900,000, 1,000,000, or more optical trapping sites. The optical trappingunits may be configured to generate at most about 1,000,000, 900,000,800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000,90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000,9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800,700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10,or fewer optical trapping sites. The optical trapping units may beconfigured to trap a number of optical trapping sites that is within arange defined by any two of the preceding values.

The optical trapping units may be configured to trap a plurality ofatoms. For instance, the optical trapping units may be configured totrap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300,400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000,7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000,70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000,600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. Theoptical trapping units may be configured to trap at most about1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000,300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000,40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000,4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100,90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The optical trappingunits may be configured to trap a number of atoms that is within a rangedefined by any two of the preceding values.

Each optical trapping site of the optical trapping units may beconfigured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or moreatoms. Each optical trapping site may be configured to trap at mostabout 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each opticaltrapping site may be configured to trap a number of atoms that is withina range defined by any two of the preceding values. Each opticaltrapping site may be configured to trap a single atom.

One or more atoms of the plurality of atoms may comprise or correspondto one more qubits (or non-classical bit), as described herein (forinstance, with respect to FIG. 4 ). For example, one atom may compriseor correspond to one qubit. Two or more atoms of the plurality may bequantum mechanically entangled. Two or more atoms of the plurality maybe quantum mechanically entangled with a coherence lifetime of at leastabout 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100μs, 200 μs, 300 μs, 400 μs, 500 μs, 600 μs, 700 μs, 800 μs, 900 μs, 1millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, 6 ms, 7 ms, 8 ms, 9 ms, 10 ms,20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms,300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or more. Two or more atomsof the plurality may be quantum mechanically entangled with a coherencelifetime of at most about 10 s, 9 s, 8 s, 7 s, 6 s, 5 s, 4 s, 3 s, 2 s,1 s, 900 ms, 800 ms, 700 ms, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms,8 ms, 7 ms, 6 ms, 5 ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 μs, 800 μs, 700 μs,600 μs, 500 μs, 400 μs, 300 μs, 200 μs, 100 μs, 90 μs, 80 μs, 70 μs, 60μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4μs, 3 μs, 2 μs, 1 μs, or less. Two or more atoms of the plurality may bequantum mechanically entangled with a coherence lifetime that is withina range defined by any two of the preceding values. One or more atoms ofthe plurality may comprise neutral atoms. One or more atoms of theplurality may comprise uncharged atoms.

One or more atoms of the plurality may comprise alkali atoms. The alkaliatoms may comprise one or more Group I elements. For example, one ormore atoms of the plurality may comprise lithium (Li) atoms, sodium (Na)atoms, potassium (K) atoms, rubidium (Rb) atoms, or caesium (Cs) atoms.One or more atoms of the plurality may comprise lithium-6 atoms,lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, orcaesium-133 atoms. One or more atoms of the plurality may comprisealkaline earth atoms. The alkaline earth atoms may comprise one or moreGroup II elements (e.g., Be or Mg). For example, one or more atoms ofthe plurality may comprise beryllium (Be) atoms, magnesium (Mg) atoms,calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One ormore atoms may comprise beryllium-9 atoms, magnesium-24 atoms,magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms,strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-134atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, orbarium-138 atoms. One or more atoms of the plurality may comprise rareearth atoms. One or more atoms of the plurality may comprise scandium(Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms,praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms,europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms,dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium(Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or moreatoms of the plurality may comprise scandium-45 atoms, yttrium-89 atoms,lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms,neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms,neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms,samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms,gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms,gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms,dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms,dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms,erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms,erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms,ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms,ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms.

The plurality of atoms may comprise a single element selected from thegroup consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. Theplurality of atoms may comprise a mixture of elements selected from thegroup consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. Theplurality of atoms may comprise a natural isotopic mixture of one ormore elements selected from the group consisting of Li, Na, K, Rb, Cs,Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise anisotopically enriched mixture of one or more elements selected from thegroup consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. Theplurality of atoms may comprise a natural isotopic mixture of one ormore elements selected from the group consisting of Sc, Y, La, Ce, Pr,Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atomsmay comprise an isotopically enriched mixture of one or more elementsselected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd,Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. Forinstance, the plurality of atoms may comprise lithium-6 atoms, lithium-7atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms,potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms,magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms,strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88atoms, barium-130 atoms, barium-132 atoms, barium-134 atoms, barium-135atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms,cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms,neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms,samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms,gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms,gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms,dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms,dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms,dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms,ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms,ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, orlutetium-176 atoms enriched to an isotopic abundance of at least about50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%,99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%,99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more.The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms,sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms,beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms,strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms,cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms,neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms,neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms,samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms,gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms,gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms,dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms,dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms,dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms,ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms,ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, orlutetium-176 atoms enriched to an isotopic abundance of at most about99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%,99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%,97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. Theplurality of atoms may comprise lithium-6 atoms, lithium-7 atoms,sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms,beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms,strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms,cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms,neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms,neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms,samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms,gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms,gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms,dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms,dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms,dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms,ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms,ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, orlutetium-176 atoms enriched to an isotopic abundance that is within arange defined by any two of the preceding values.

The system 200 may comprise one or more electromagnetic delivery units220. The electromagnetic delivery units may comprise any electromagneticdelivery unit described herein, such as an electromagnetic delivery unitdescribed herein with respect to FIG. 4 . The electromagnetic deliveryunits may be configured to apply electromagnetic energy to one or moreatoms of the plurality of atoms. Applying the electromagnetic energy mayinduce the atoms to adopt one or more superposition states of a firstatomic state and a second atomic state that is different from the firstatomic state.

The first atomic state may comprise a first single-qubit state. Thesecond atomic state may comprise a second single-qubit state. The firstatomic state or second atomic state may be elevated in energy withrespect to a ground atomic state of the atoms. The first atomic state orsecond atomic state may be equal in energy with respect to the groundatomic state of the atoms.

The first atomic state may comprise a first hyperfine electronic stateand the second atomic state may comprise a second hyperfine electronicstate that is different from the first hyperfine electronic state. Forinstance, the first and second atomic states may comprise first andsecond hyperfine states on a multiplet manifold, such as a tripletmanifold. The first and second atomic states may comprise first andsecond hyperfine states, respectively, on a ³P₁ or ³P₂ manifold. Thefirst and second atomic states may comprise first and second hyperfinestates, respectively, on a ³P₁ or ³P₂ manifold of any atom describedherein, such as a strontium-87 ³P₁ manifold or a strontium-87 ³P₂manifold.

FIG. 9 shows an example of a qubit comprising a ³P₂ state ofstrontium-87. The left panel of FIG. 9 shows the rich energy levelstructure of the ³P₂ state of strontium-87. The right panel of FIG. 9shows a potential qubit transition within the ³P₂ state of strontium-87which is insensitive (to first order) to changes in magnetic fieldaround 70 Gauss.

The first atomic state may comprise a first nuclear spin state and thesecond atomic state may comprise a second nuclear spin state that isdifferent from the first nuclear spin state. The first and second atomicstates may comprise first and second nuclear spin states, respectively,of a nucleus comprising a nuclear spin greater than-½. The first andsecond atomic states may comprise first and second nuclear spin states,respectively, of a spin-1, spin-3/2, spin-2, spin-5/2, spin-3, spin-7/2,spin-4, or spin-9/2 nucleus. The first and second atomic states maycomprise first and second nuclear spin states, respectively, of any atomdescribed herein, such as first and second spin states of strontium-87.

For first and second nuclear spin states associated with a nucleuscomprising a nuclear spin greater than ½ (such as a spin-1, spin-3/2,spin-2, spin-5/2, spin-3, spin-7/2, spin-4, or spin-9/2 nucleus),transitions between the first and second nuclear spin states may beaccompanied by transitions between other spin states on the nuclear spinmanifold. For instance, for a spin-9/2 nucleus in the presence of auniform magnetic field, all of the nuclear spin levels may be separatedby equal energy. Thus, a transition (such as a Raman transition)designed to transfer atoms from, for instance, an m_(N)=9/2 spin stateto an m_(N)=7/2 spin state, may also drive m_(N)=7/2 to m_(N)=5/2,m_(N)=5/2 to m_(N)=3/2, m_(N)=3/2 to m_(N)=1/2, m_(N)=1/2 to m_(N)=−1/2,m_(N)=−1/2 to m_(N)=−3/2, m_(N)=−3/2 to m_(N)=−5/2, m_(N)=−5/2 tom_(N)=−7/2, and m_(N)=−7/2 to m_(N)=−9/2, where m_(N) is the nuclearspin state. Similarly, a transition (such as a Raman transition)designed to transfer atoms from, for instance, an m_(N)=9/2 spin stateto an m_(N)=5/2 spin state, may also drive m_(N)=7/2 to m_(N)=3/2,m_(N)=5/2 to m_(N)=1/2, m_(N)=3/2 to m_(N)=−1/2, m_(N)=1/2 tom_(N)=−3/2, m_(N)=−1/2 to m_(N)=−5/2, m_(N)=−3/2 to m_(N)=−7/2, andm_(N)=−5/2 to m_(N)=−9/2. Such a transition may thus not be selectivefor inducing transitions between particular spin states on the nuclearspin manifold. It may be desirable to instead implement selectivetransitions between particular first and second spins states on thenuclear spin manifold. This may be accomplished by providing light froma light source that provides an AC Stark shift and pushes neighboringnuclear spin states out of resonance with a transition between thedesired transition between the first and second nuclear spin states. Forinstance, if a transition from first and second nuclear spin stateshaving m_(N)=−9/2 and m_(N)=−7/2 is desired, the light may provide an ACStark shift to the m_(N)=−5/2 spin state, thereby greatly reducingtransitions between the m_(N)=−7/2 and m_(N)=−5/2 states. Similarly, ifa transition from first and second nuclear spin states having m_(N)=−9/2and m_(N)=−5/2 is desired, the light may provide an AC Stark shift tothe m_(N)=−1/2 spin state, thereby greatly reducing transitions betweenthe m_(N)=−5/2 and m_(N)=−1/2 states. This may effectively create atwo-level subsystem within the nuclear spin manifold that is decoupledfrom the remainder of the nuclear spin manifold, greatly simplifying thedynamics of the qubit systems. It may be advantageous to use nuclearspin states near the edge of the nuclear spin manifold (e.g., m_(N)=−9/2and m_(N)=−7/2, m_(N)=7/2 and m_(N)=9/2, m_(N)=−9/2 and m_(N)=−5/2, orm_(N)=5/2 and m_(N)=9/2 for a spin-9/2 nucleus) such that only one ACStark shift is required. Alternatively, nuclear spin states farther fromthe edge of the nuclear spin manifold (e.g., m_(N)=−5/2 and m_(N)=−3/2or m_(N)=−5/2 and m_(N)=−1/2) may be used and two AC Stark shifts may beimplemented (e.g., at m_(N)=−7/2 and m_(N)=−1/2 or m_(N)=−9/2 andm_(N)=3/2).

Qubits based on nuclear spin states in the electronic ground state mayallow exploitation of long-lived metastable excited electronic states(such as a ³P₀ state in strontium-87) for qubit storage. Atoms may beselectively transferred into such a state to reduce cross-talk or toimprove gate or detection fidelity. Such a storage or shelving processmay be atom-selective using the SLMs or AODs described herein.

The system 200 may comprise one or more readout optical units 230. Thereadout optical units may be configured to perform one or moremeasurements of the one or more superposition states to obtain thenon-classical computation. The readout optical units may comprise one ormore optical detectors. The detectors may comprise one or morephotomultiplier tubes (PMTs), photodiodes, avalanche diodes,single-photon avalanche diodes, single-photon avalanche diode arrays,phototransistors, reverse-biased light emitting diodes (LEDs), chargecoupled devices (CCDs), or complementary metal oxide semiconductor(CMOS) cameras. The optical detectors may comprise one or morefluorescence detectors. The readout optical unit may comprise one ormore objectives, such as one or more objective having a numericalaperture (NA) of at least about 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4,0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, or more.The objective may have an NA of at most about 1, 0.95, 0.9, 0.85, 0.8,0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15,0.1, or less. The objective may have an NA that is within a rangedefined by any two of the preceding values.

The system 200 may comprise one or more vacuum units 240. The one ormore vacuum units may comprise one or more vacuum pumps. The vacuumunits may comprise one or more roughing vacuum pumps, such as one ormore rotary pumps, rotary vane pumps, rotary piston pumps, diaphragmpumps, piston pumps, reciprocating piston pumps, scroll pumps, or screwpumps. The one or more roughing vacuum pumps may comprise one or morewet (for instance, oil-sealed) or dry roughing vacuum pumps. The vacuumunits may comprise one or more high-vacuum pumps, such as one or morecryosorption pumps, diffusion pumps, turbomolecular pumps, moleculardrag pumps, turbo-drag hybrid pumps, cryogenic pumps, ions pumps, orgetter pumps.

The vacuum units may comprise any combination of vacuum pumps describedherein. For instance, the vacuum units may comprise one or more roughingpumps (such as a scroll pump) configured to provide a first stage ofrough vacuum pumping. The roughing vacuum pumps may be configured topump gases out of the system 200 to achieve a low vacuum pressurecondition. For instance, the roughing pumps may be configured to pumpgases out of the system 200 to achieve a low vacuum pressure of at mostabout 10³ Pascals (Pa). The vacuum units may further comprise one ormore high-vacuum pumps (such as one or more ion pumps, getter pumps, orboth) configured to provide a second stage of high vacuum pumping orultra-high vacuum pumping. The high-vacuum pumps may be configured topump gases out of the system 200 to achieve a high vacuum pressure of atmost about 10⁻³ Pa or an ultra-high vacuum pressure of at most about10⁻⁶ Pa once the system 200 has reached the low vacuum pressurecondition provided by the one or more roughing pumps.

The vacuum units may be configured to maintain the system 200 at apressure of at most about 10⁻⁶ Pa, 9×10⁻⁷ Pa, 8×10⁻⁷ Pa, 7×10⁻⁷ Pa,6×10⁻⁷ Pa, 5×10⁻⁷ Pa, 4×10⁻⁷ Pa, 3×10⁻⁷ Pa, 2×10⁻⁷ Pa, 10⁻⁷ Pa, 9×10⁻⁸Pa, 8×10⁻⁸ Pa, 7×10⁻⁸ Pa, 6×10⁻⁸ Pa, 5×10⁻⁸ Pa, 4×10⁻⁸ Pa, 3×10⁻⁸ Pa,2×10⁻⁸ Pa, 10⁻⁸ Pa, 9×10⁻⁹ Pa, 8×10⁻⁹ Pa, 7×10⁻⁹ Pa, 6×10⁻⁹ Pa, 5×10⁻⁹Pa, 4×10⁻⁹ Pa, 3×10⁻⁹ Pa, 2×10⁻⁹ Pa, 10⁻⁹ Pa, 9×10⁻¹⁰ Pa, 8×10⁻¹⁰ Pa,7×10⁻¹⁰ Pa, 6×10⁻¹⁰ Pa, 5×10⁻¹⁰ Pa, 4×10⁻¹⁰ Pa, 3×10⁻¹⁰ Pa, 2×10⁻¹⁰ Pa,10⁻¹⁰ Pa, 9×10⁻¹¹ Pa, 8×10⁻¹¹ Pa, 7×10⁻¹¹ Pa, 6×10⁻¹¹ Pa, 5×10⁻¹¹ Pa,4×10⁻¹¹ Pa, 3×10⁻¹¹ Pa, 2×10⁻¹¹ Pa, 10⁻¹¹ Pa, 9×10⁻¹² Pa, 8×10⁻¹² Pa,7×10⁻¹² Pa, 6×10⁻¹² Pa, 5×10⁻¹² Pa, 4×10⁻¹² Pa, 3×10⁻¹² Pa, 2×10⁻¹² Pa,10⁻¹² Pa, or lower. The vacuum units may be configured to maintain thesystem 200 at a pressure of at least about 10⁻¹² Pa, 2×10⁻¹² Pa, 3×10⁻¹²Pa, 4×10⁻¹² Pa, 5×10⁻¹² Pa, 6×10⁻¹² Pa, 7×10⁻¹² Pa, 8×10⁻¹² Pa, 9×10⁻¹²Pa, 10⁻¹¹ Pa, 2×10⁻¹¹ Pa, 3×10⁻¹¹ Pa, 4×10⁻¹¹ Pa, 5×10⁻¹¹ Pa, 6×10⁻¹¹Pa, 7×10⁻¹¹ Pa, 8×10⁻¹¹ Pa, 9×10⁻¹¹ Pa, 10⁻¹⁰ Pa, 2×10⁻¹⁰ Pa, 3×10⁻¹⁰Pa, 4×10⁻¹⁰ Pa, 5×10⁻¹⁰ Pa, 6×10⁻¹⁰ Pa, 7×10⁻¹⁰ Pa, 8×10⁻¹⁰ Pa, 9×10⁻¹⁰Pa, 10⁻⁹ Pa, 2×10⁻⁹ Pa, 3×10⁻⁹ Pa, 4×10⁻⁹ Pa, 5×10⁻⁹ Pa, 6×10⁻⁹ Pa,7×10⁻⁹ Pa, 8×10⁻⁹ Pa, 9×10⁻⁹ Pa, 10⁻⁸ Pa, 2×10⁻⁸ Pa, 3×10⁻⁸ Pa, 4×10⁻⁸Pa, 5×10⁻⁸ Pa, 6×10⁻⁸ Pa, 7×10⁻⁸ Pa, 8×10⁻⁸ Pa, 9×10⁻⁸ Pa, 10⁻⁷ Pa,2×10⁻⁷ Pa, 3×10⁻⁷ Pa, 4×10⁻⁷ Pa, 5×10⁻⁷ Pa, 6×10⁻⁷ Pa, 7×10⁻⁷ Pa, 8×10⁻⁷Pa, 9×10⁻⁷ Pa, 10⁻⁶ Pa, or higher. The vacuum units may be configured tomaintain the system 200 at a pressure that is within a range defined byany two of the preceding values.

The system 200 may comprise one or more state preparation units 250. Thestate preparation units may comprise any state preparation unitdescribed herein, such as a state preparation unit described herein withrespect to FIG. 5 . The state preparation units may be configured toprepare a state of the plurality of atoms.

The system 200 may comprise one or more atom reservoirs 260. The atomreservoirs may be configured to supply one or more replacement atoms toreplace one or more atoms at one or more optical trapping sites uponloss of the atoms from the optical trapping sites.

The system 200 may comprise one or more atom movement units 270. Theatom movement units may be configured to move the one or morereplacement atoms to the one or more optical trapping sites. Forinstance, the one or more atom movement units may comprise one or moreelectrically tunable lenses, acousto-optic deflectors (AODs), or spatiallight modulators (SLMs).

The system 200 may comprise one or more entanglement units 280. Theentanglement units may be configured to quantum mechanically entangle atleast a first atom of the plurality of atoms with at least a second atomof the plurality of atoms. The first or second atom may be in asuperposition state at the time of quantum mechanical entanglement.Alternatively or in combination, the first or second atom may not be ina superposition state at the time of quantum mechanical entanglement.The first atom and the second atom may be quantum mechanically entangledthrough one or more magnetic dipole interactions, electric dipoleinteractions, or induced electric dipole interactions. The entanglementunits may be configured to quantum mechanically entangle any number ofatoms described herein.

The entanglement units may comprise one or more Rydberg excitationunits. The Rydberg excitation units may be configured to electronicallyexcite the subset of the one or more atoms in the one or moresuperposition states to a Rydberg state, thereby forming one or moreRydberg atoms. Alternatively or in combination, the Rydberg excitationunits may be configured to continuously drive the one or more atoms inthe one or more superposition states via a transition to a Rydbergstate, thereby forming one or more dressed Rydberg atoms. The Rydbergexcitation units may be configured to induce one or more quantummechanical entanglements between the Rydberg atoms or dressed Rydbergatoms and the second atom. The second atom may be located at a distanceof at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm,700 nm, 800 nm, 900 nm, 1 micrometer (μm), 2 μm, 3 μm, 4 μm, 5 μm, 6 μm,7 μm, 8 μm, 9 μm, 10 μm, or more from the Rydberg atoms or dressedRydberg atoms. The second atom may be located at a distance of at mostabout 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less fromthe Rydberg atoms or dressed Rydberg atoms. The second atom may belocated at a distance from the Rydberg atoms or dressed Rydberg atomsthat is within a range defined by any two of the preceding values. TheRydberg excitation units may be configured to allow the Rydberg atoms ordressed Rydberg atoms to relax to a lower-energy atomic state, therebyforming one or more two-qubit units. The Rydberg excitation units may beconfigured to induce the Rydberg atoms or dressed Rydberg atoms to relaxto a lower-energy atomic state, thereby forming one or more two-qubitunits. The Rydberg excitation units may be configured to drive theRydberg atoms or dressed Rydberg atoms to a lower-energy atomic state,thereby forming one or more two-qubit units. For instance, the Rydbergexcitation units may be configured to apply electromagnetic radiation(such as RF radiation or optical radiation) to drive the Rydberg atomsor dressed Rydberg atoms to a lower-energy atomic state. The Rydbergexcitation units may be configured to induce any number of quantummechanical entanglements between any number of atoms of the plurality ofatoms.

The Rydberg excitation units may comprise one or more light sources(such as any light source described herein) configured to emit lighthaving one or more ultraviolet (UV) wavelengths. The UV wavelengths maybe selected to correspond to a wavelength that forms the Rydberg atomsor dressed Rydberg atoms. For instance, the light may comprise one ormore wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. Thelight may comprise one or more wavelengths of at most about 400 nm, 390nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210nm, 200 nm, or less. The light may comprise one or more wavelengths thatare within a range defined by any two of the preceding values. Forinstance, the light may comprise one or more wavelengths that are withina range from 300 nm to 400 nm.

The Rydberg excitation units may be configured to induce a two-photontransition to generate an entanglement. The Rydberg excitation units maybe configured to induce a two-photon transition to generate anentanglement between two atoms. The Rydberg excitation units may beconfigured to selectively induce a two-photon transition to selectivelygenerate an entanglement between two atoms. For instance, the Rydbergexcitation units may be configured to direct electromagnetic energy(such as optical energy) to particular optical trapping sites toselectively induce a two-photon transition to selectively generate theentanglement between the two atoms. The two atoms may be trapped innearby optical trapping sites. For instance, the two atoms may betrapped in adjacent optical trapping sites. The two-photon transitionmay be induced using first and second light from first and second lightsources, respectively. The first and second light sources may eachcomprise any light source described herein (such as any laser describedherein). The first light source may be the same or similar to a lightsource used to perform a single-qubit operation described herein.Alternatively, different light sources may be used to perform asingle-qubit operation and to induce a two-photon transition to generatean entanglement. The first light source may emit light comprising one ormore wavelengths in the visible region of the optical spectrum (e.g.,within a range from 400 nm to 800 nm or from 650 nm to 700 nm). Thesecond light source may emit light comprising one or more wavelengths inthe ultraviolet region of the optical spectrum (e.g., within a rangefrom 200 nm to 400 nm or from 300 nm to 350 nm). The first and secondlight sources may emit light having substantially equal and oppositespatially-dependent frequency shifts.

The Rydberg atoms or dressed Rydberg atoms may comprise a Rydberg statethat may have sufficiently strong interatomic interactions with nearbyatoms (such as nearby atoms trapped in nearby optical trapping sites) toenable the implementation of two-qubit operations. The Rydberg statesmay comprise a principal quantum number of at least about 50, 60, 70,80, 90, 100, or more. The Rydberg states may comprise a principalquantum number of at most about 100, 90, 80, 70, 60, 50, or less. TheRydberg states may comprise a principal quantum number that is within arange defined by any two of the preceding values. The Rydberg states mayinteract with nearby atoms through van der Waals interactions. The vander Waals interactions may shift atomic energy levels of the atoms.

State selective excitation of atoms to Rydberg levels may enable theimplementation of two-qubit operations. Two-photon transitions may beused to excite atoms from a ground state (such as a ¹S₀ ground state) toa Rydberg state (such as an n³S₁ state, wherein n is a principal quantumnumber described herein). State selectivity may be accomplished by acombination of laser polarization and spectral selectivity. Thetwo-photon transitions may be implemented using first and second lasersources, as described herein. The first laser source may emitpi-polarized light, which may not change the projection of atomicangular momentum along a magnetic field. The second laser may emitcircularly polarized light, which may change the projection of atomicangular momentum along the magnetic field by one unit. The first andsecond qubit levels may be excited to Rydberg level using thispolarization. However, the Rydberg levels may be more sensitive tomagnetic fields than the ground state so large splittings (for instance,on the order of 100s of MHz) may be readily obtained. This spectralselectivity may allow state selective excitation to Rydberg levels.

Two-qubit operations may rely on energy shifts of levels due to Van derWaals interactions described herein. Such shifts may either prevent theexcitation of one atom conditional on the state of the other or changethe coherent dynamics of excitation of the two-atom system to enact atwo-qubit operation. In some cases, “dressed states” may be generatedunder continuous driving to enact two-qubit operations without requiringfull excitation to a Rydberg level (for instance, as described inhttps://arxiv.org/abs/1605.05207, which is incorporated herein byreference in its entirety for all purposes).

Cloud Computing

The system 200 may be operatively coupled to a digital computerdescribed herein (such as a digital computer described herein withrespect to FIG. 1 ) over a network described herein (such as a networkdescribed herein with respect to FIG. 1 ). The network may comprise acloud computing network.

Optical Trapping Units

FIG. 3A shows an example of an optical trapping unit 210. The opticaltrapping unit may be configured to generate a plurality 211 of spatiallydistinct optical trapping sites, as described herein. For instance, asshown in FIG. 3B, the optical trapping unit may be configured togenerate a first optical trapping site 211 a, second optical trappingsite 211 b, third optical trapping site 211 c, fourth optical trappingsite 211 d, fifth optical trapping site 211 e, sixth optical trappingsite 211 f, seventh optical trapping site 211 g, eighth optical trappingsite 211 h, and ninth optical trapping site 211 i, as depicted in FIG.3A. The plurality of spatially distinct optical trapping sites may beconfigured to trap a plurality of atoms, such as first atom 212 a,second atom 212 b, third atom 212 c, and fourth atom 212 d, as depictedin FIG. 3A. As depicted in FIG. 3B, each optical trapping site may beconfigured to trap a single atom. As depicted in FIG. 3B, some of theoptical trapping sites may be empty (i.e., not trap an atom).

As shown in FIG. 3B, the plurality of optical trapping sites maycomprise a two-dimensional (2D) array. The 2D array may be perpendicularto the optical axis of optical components of the optical trapping unitdepicted in FIG. 3A. Alternatively, the plurality of optical trappingsites may comprise a one-dimensional (1D) array or a three-dimensional(3D) array.

Although depicted as comprising nine optical trapping sites filled byfour atoms in FIG. 3B, the optical trapping unit 210 may be configuredto generate any number of spatially distinct optical trapping sitesdescribed herein and may be configured to trap any number of atomsdescribed herein.

Each optical trapping site of the plurality of optical trapping sitesmay be spatially separated from each other optical trapping site by adistance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700nm, 800 nm, 900 nm, 1 μm, 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9μm, 10 μm, or more. Each optical trapping site may be spatiallyseparated from each other optical trapping site by a distance of at mostabout 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less.Each optical trapping site maybe spatially separated from each otheroptical trapping site by a distance that is within a range defined byany two of the preceding values.

The optical trapping sites may comprise one or more optical tweezers.Optical tweezers may comprise one or more focused laser beams to providean attractive or repulsive force to hold or move the one or more atoms.The beam waist of the focused laser beams may comprise a strong electricfield gradient. The atoms may be attracted or repelled along theelectric field gradient to the center of the laser beam, which maycontain the strongest electric field. The optical trapping sites maycomprise one or more optical lattice sites of one or more opticallattices. The optical trapping sites may comprise one or more opticallattice sites of one or more one-dimensional (1D) optical lattices,two-dimensional (2D) optical lattices, or three-dimensional (3D) opticallattices. For instance, the optical trapping sites may comprise one ormore optical lattice sites of a 2D optical lattice, as depicted in FIG.3B.

The optical lattices may be generated by interfering counter-propagatinglight (such as counter-propagating laser light) to generate a standingwave pattern having a periodic succession of intensity minima and maximaalong a particular direction. A 1D optical lattice may be generated byinterfering a single pair of counter-propagating light beams. A 2Doptical lattice may be generated by interfering two pairs ofcounter-propagating light beams. A 3D optical lattice may be generatedby interfering three pairs of counter-propagating lights beams. Thelight beams may be generated by different light sources or by the samelight source. Therefore, an optical lattice may be generated by at leastabout 1, 2, 3, 4, 5, 6, or more light sources or at most about 6, 5, 4,3, 2, or 1 light sources.

Returning to the description of FIG. 3A, the optical trapping unit maycomprise one or more light sources configured to emit light to generatethe plurality of optical trapping sites as described herein. Forinstance, the optical trapping unit may comprise a single light source213, as depicted in FIG. 3A. Though depicted as comprising a singlelight source in FIG. 3A, the optical trapping unit may comprise anynumber of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8,9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3,2, or 1 light sources. The light sources may comprise one or morelasers.

The lasers may comprise one or more continuous wave lasers. The lasersmay comprise one or more pulsed lasers. The lasers may comprise one ormore gas lasers, such as one or more helium-neon (HeNe) lasers, argon(Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N₂)lasers, carbon dioxide (CO₂) lasers, carbon monoxide (CO) lasers,transversely excited atmospheric (TEA) lasers, or excimer lasers. Forinstance, the lasers may comprise one or more argon dimer (Ar₂) excimerlasers, krypton dimer (Kr₂) excimer lasers, fluorine dimer (F₂) excimerlasers, xenon dimer (Xe₂) excimer lasers, argon fluoride (ArF) excimerlasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF)excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride(XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The lasermay comprise one or more dye lasers.

The lasers may comprise one or more metal-vapor lasers, such as one ormore helium-cadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg)metal-vapor lasers, helium-selenium (HeSe) metal-vapor lasers,helium-silver (HeAg) metal-vapor lasers, strontium (Sr) metal-vaporlasers, neon-copper (NeCu) metal-vapor lasers, copper (Cu) metal-vaporlasers, gold (Au) metal-vapor lasers, manganese (Mn) metal-vapor laser,or manganese chloride (MnCl₂) metal-vapor lasers.

The lasers may comprise one or more solid-state lasers, such as one ormore ruby lasers, metal-doped crystal lasers, or metal-doped fiberlasers. For instance, the lasers may comprise one or moreneodymium-doped yttrium aluminum garnet (Nd:YAG) lasers,neodymium/chromium doped yttrium aluminum garnet (Nd/Cr:YAG) lasers,erbium-doped yttrium aluminum garnet (Er:YAG) lasers, neodymium-dopedyttrium lithium fluoride (Nd:YLF) lasers, neodymium-doped yttriumorthovanadate (ND:YVO₄) lasers, neodymium-doped yttrium calciumoxoborate (Nd:YCOB) lasers, neodymium glass (Nd:glass) lasers, titaniumsapphire (Ti:sapphire) lasers, thulium-doped ytrrium aluminum garnet(Tm:YAG) lasers, ytterbium-doped ytrrium aluminum garnet (Yb:YAG)lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrriumaluminum garnet (Ho:YAG) lasers, chromium-doped zinc selenide (Cr:ZnSe)lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF)lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF)lasers, erbium-doped glass (Er:glass) lasers, erbium-ytterbium-codopedglass (Er/Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF₂)lasers, or samarium-doped calcium fluoride (Sm:CaF₂) lasers.

The lasers may comprise one or more semiconductor lasers or diodelasers, such as one or more gallium nitride (GaN) lasers, indium galliumnitride (InGaN) lasers, aluminum gallium indium phosphide (AlGaInP)lasers, aluminum gallium arsenide (AlGaAs) lasers, indium galliumarsenic phosphide (InGaAsP) lasers, vertical cavity surface emittinglasers (VCSELs), or quantum cascade lasers.

The lasers may emit continuous wave laser light. The lasers may emitpulsed laser light. The lasers may have a pulse length of at least about1 femtoseconds (fs), 2 fs, 3 fs, 4 fs, 5 fs, 6 fs, 7 fs, 8 fs, 9 fs, 10fs, 20 fs, 30 fs, 40 fs, 50 fs, 60 fs, 70 fs, 80 fs, 90 fs, 100 fs, 200fs, 300 fs, 400 fs, 500 fs, 600 fs, 700 fs, 800 fs, 900 fs, 1 picosecond(ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps,400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 nanosecond (ns), 2 ns,3 ns, 4 ns, 5 ns, 6 ns, 7 ns, 8 ns, 9 ns, 10 ns, 20 ns, 30 ns, 40 ns, 50ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns,600 ns, 700 ns, 800 ns, 900 ns, 1,000 ns, or more. The lasers may have apulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns,500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3ns, 2 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps,200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps,10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or less. The lasers may have a pulselength that is within a range defined by any two of the precedingvalues.

The lasers may have a repetition rate of at least about 1 hertz (Hz), 2Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz,500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz,4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz,400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz),2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz,30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz,1,000 MHz, or more. The lasers may have a repetition rate of at mostabout 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz,300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz,300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, orless. The lasers may have a repetition rate that is within a rangedefined by any two of the preceding values.

The lasers may emit light having a pulse energy of at least about 1nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ,20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ,300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule(μJ), 2 μJ, 3 μJ, 4 μJ, 5 μJ, 6 μJ, 7 μJ, 8 μJ, 9 μJ, 10 μJ, 20 μJ, 30μJ, 40 μJ, 50 μJ, 60 μJ, 70 μJ, 80 μJ, 90 μJ, 100 μJ, 200 μJ, 300 μJ,400 μJ, 500 μJ, 600 μJ, 700 μJ, 800 μJ, 900 μJ, a least 1 millijoule(mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ,400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), ormore. The lasers may emit light having a pulse energy of at most about 1J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ,8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 μJ, 800 μJ, 700 μJ,600 μJ, 500 μJ, 400 μJ, 300 μJ, 200 μJ, 100 μJ, 90 μJ, 80 μJ, 70 μJ, 60μJ, 50 μJ, 40 μJ, 30 μJ, 20 μJ, 10 μJ, 9 μJ, 8 μJ, 7 μJ, 6 μJ, 5 μJ, 4μJ, 3 μJ, 2 μJ, 1 μJ, 900 nJ, 800 nJ, 700 nJ, 600 nJ, 500 nJ, 400 nJ,300 nJ, 200 nJ, 100 nJ, 90 nJ, 80 nJ, 70 nJ, 60 nJ, 50 nJ, 40 nJ, 30 nJ,20 nJ, 10 nJ, 9 nJ, 8 nJ, 7 nJ, 6 nJ, 5 nJ, 4 nJ, 3 nJ, 2 nJ, 1 nJ, orless. The lasers may emit light having a pulse energy that is within arange defined by any two of the preceding values.

The lasers may emit light having an average power of at least about 1microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW,20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW,300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt(mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW,400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800 W, 900 W,1,000 W, or more. The lasers may emit light having an average power ofat most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W,200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W,8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW,500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW,200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW,10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or more.The lasers may emit light having a power that is within a range definedby any two of the preceding values.

The lasers may emit light comprising one or more wavelengths in theultraviolet (UV), visible, or infrared (IR) portions of theelectromagnetic spectrum. The lasers may emit light comprising one ormore wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm,1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm,1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm,1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm,1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm,1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370 nm, 1,380 nm,1,390 nm, 1,400 nm, or more. The lasers may emit light comprising one ormore wavelengths of at most about 1,400 nm, 1,390 nm, 1,380 nm, 1,370 n,1,360 nm, 1,350 nm, 1,340 nm, 1,330 nm, 1,320 nm, 1,310 nm, 1,300 nm,1,290 nm, 1,280 nm, 1,270 n, 1,260 nm, 1,250 nm, 1,240 nm, 1,230 nm,1,220 nm, 1,210 nm, 1,200 nm, 1,190 nm, 1,180 nm, 1,170 n, 1,160 nm,1,150 nm, 1,140 nm, 1,130 nm, 1,120 nm, 1,110 nm, 1,100 nm, 1,090 nm,1,080 nm, 1,070 n, 1,060 nm, 1,050 nm, 1,040 nm, 1,030 nm, 1,020 nm,1,010 nm, 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210nm, 200 nm. The lasers may emit light comprising one or more wavelengthsthat are within a range defined by any two of the preceding values.

The lasers may emit light having a bandwidth of at least about 1×10⁻¹⁵nm, 2×10⁻¹⁵ nm, 3×10⁻¹⁵ nm, 4×10⁻¹⁵ nm, 5×10⁻¹⁵ nm, 6×10⁻¹⁵ nm, 7×10⁻¹⁵nm, 8×10⁻¹⁵ nm, 9×10⁻¹⁵ nm, 1×10⁻¹⁴ nm, 2×10⁻¹⁴ nm, 3×10⁻¹⁴ nm, 4×10⁻¹⁴nm, 5×10⁻¹⁴ nm, 6×10⁻¹⁴ nm, 7×10⁻¹⁴ nm, 8×10⁻¹⁴ nm, 9×10⁻¹⁴ nm, 1×10⁻¹³nm, 2×10⁻¹³ nm, 3×10⁻¹³ nm, 4×10⁻¹³ nm, 5×10⁻¹³ nm, 6×10⁻¹³ nm, 7×10⁻¹³nm, 8×10⁻¹³ nm, 9×10⁻¹³ nm, 1×10⁻¹² nm, 2×10⁻¹² nm, 3×10⁻¹² nm, 4×10⁻¹²nm, 5×10⁻¹² nm, 6×10⁻¹² nm, 7×10⁻¹² nm, 8×10⁻¹² nm, 9×10⁻¹² nm, 1×10⁻¹¹nm, 2×10⁻¹¹ nm, 3×10⁻¹¹ nm, 4×10⁻¹¹ nm, 5×10⁻¹¹ nm, 6×10⁻¹¹ nm, 7×10⁻¹¹nm, 8×10⁻¹¹ nm, 9×10⁻¹¹ nm, 1×10⁻¹⁰ nm, 2×10⁻¹⁰ nm, 3×10⁻¹⁰ nm, 4×10⁻¹⁰nm, 5×10⁻¹⁰ nm, 6×10⁻¹⁰ nm, 7×10⁻¹⁰ nm, 8×10⁻¹⁰ nm, 9×10⁻¹⁰ nm, 1×10⁻⁹nm, 2×10⁻⁹ nm, 3×10⁻⁹ nm, 4×10⁻⁹ nm, 5×10⁻⁹ nm, 6×10⁻⁹ nm, 7×10⁻⁹ nm,8×10⁻⁹ nm, 9×10⁻⁹ nm, 1×10⁻⁸ nm, 2×10⁻⁸ nm, 3×10⁻⁸ nm, 4×10⁻⁸ nm, 5×10⁻⁸nm, 6×10⁻⁸ nm, 7×10⁻⁸ nm, 8×10⁻⁸ nm, 9×10⁻⁸ nm, 1×10⁻⁷ nm, 2×10⁻⁷ nm,3×10⁻⁷ nm, 4×10⁻⁷ nm, 5×10⁻⁷ nm, 6×10⁻⁷ nm, 7×10⁻⁷ nm, 8×10⁻⁷ nm, 9×10⁻⁷nm, 1×10⁻⁶ nm, 2×10⁻⁶ nm, 3×10⁻⁶ nm, 4×10⁻⁶ nm, 5×10⁻⁶ nm, 6×10⁻⁶ nm,7×10⁻⁶ nm, 8×10⁻⁶ nm, 9×10⁻⁶ nm, 1×10⁻⁵ nm, 2×10⁻⁵ nm, 3×10⁻⁵ nm, 4×10⁻⁵nm, 5×10⁻⁵ nm, 6×10⁻⁵ nm, 7×10⁻⁵ nm, 8×10⁻⁵ nm, 9×10⁻⁵ nm, 1×10⁻⁴ nm,2×10⁻⁴ nm, 3×10⁻⁴ nm, 4×10⁻⁴ nm, 5×10⁻⁴ nm, 6×10⁻⁴ nm, 7×10⁻⁴ nm, 8×10⁻⁴nm, 9×10⁻⁴ nm, 1×10⁻³ nm, or more. The lasers may emit light having abandwidth of at most about 1×10⁻³ nm, 9×10⁻⁴ nm, 8×10⁻⁴ nm, 7×10⁻⁴ nm,6×10⁻⁴ nm, 5×10⁻⁴ nm, 4×10⁻⁴ nm, 3×10⁻⁴ nm, 2×10⁻⁴ nm, 1×10⁻⁴ nm, 9×10⁻⁵nm, 8×10⁻⁵ nm, 7×10⁻⁵ nm, 6×10⁻⁵ nm, 5×10⁻⁵ nm, 4×10⁻⁵ nm, 3×10⁻⁵ nm,2×10⁻⁵ nm, 1×10⁻⁵ nm, 9×10⁻⁶ nm, 8×10⁻⁶ nm, 7×10⁻⁶ nm, 6×10⁻⁶ nm, 5×10⁻⁶nm, 4×10⁻⁶ nm, 3×10⁻⁶ nm, 2×10⁻⁶ nm, 1×10⁻⁶ nm, 9×10⁻⁷ nm, 8×10⁻⁷ nm,7×10⁻⁷ nm, 6×10⁻⁷ nm, 5×10⁻⁷ nm, 4×10⁻⁷ nm, 3×10⁻⁷ nm, 2×10⁻⁷ nm, 1×10⁻⁷nm, 9×10⁻⁸ nm, 8×10⁻⁸ nm, 7×10⁻⁸ nm, 6×10⁻⁸ nm, 5×10⁻⁸ nm, 4×10⁻⁸ nm,3×10⁻⁸ nm, 2×10⁻⁸ nm, 1×10⁻⁸ nm, 9×10⁻⁹ nm, 8×10⁻⁹ nm, 7×10⁻⁹ nm, 6×10⁻⁹nm, 5×10⁻⁹ nm, 4×10⁻⁹ nm, 3×10⁻⁹ nm, 2×10⁻⁹ nm, 1×10⁻⁹ nm, 9×10⁻¹⁰ nm,8×10⁻¹⁰ nm, 7×10⁻¹⁰ nm, 6×10⁻¹⁰ nm, 5×10⁻¹⁰ nm, 4×10⁻¹⁰ nm, 3×10⁻¹⁰ nm,2×10⁻¹⁰ nm, 1×10⁻¹⁰ nm, 9×10⁻¹¹ nm, 8×10⁻¹¹ nm, 7×10⁻¹¹ nm, 6×10⁻¹¹ nm,5×10⁻¹¹ nm, 4×10⁻¹¹ nm, 3×10⁻¹¹ nm, 2×10⁻¹¹ nm, 1×10⁻¹¹ nm, 9×10⁻¹² nm,8×10⁻¹² nm, 7×10⁻¹² nm, 6×10⁻¹² nm, 5×10⁻¹² nm, 4×10⁻¹² nm, 3×10⁻¹² nm,2×10⁻¹² nm, 1×10⁻¹² nm, 9×10⁻¹³ nm, 8×10⁻¹³ nm, 7×10⁻¹³ nm, 6×10⁻¹³ nm,5×10⁻¹³ nm, 4×10⁻¹³ nm, 3×10⁻¹³ nm, 2×10⁻¹³ nm, 1×10⁻¹³ nm, 9×10⁻¹⁴ nm,8×10⁻¹⁴ nm, 7×10⁻¹⁴ nm, 6×10⁻¹⁴ nm, 5×10⁻¹⁴ nm, 4×10⁻¹⁴ nm, 3×10⁻¹⁴ nm,2×10⁻¹⁴ nm, 1×10⁻¹⁴ nm, 9×10⁻¹⁵ nm, 8×10⁻¹⁵ nm, 7×10⁻¹⁵ nm, 6×10⁻¹⁵ nm,5×10⁻¹⁵ nm, 4×10⁻¹⁵ nm, 3×10⁻¹⁵ nm, 2×10⁻¹⁵ nm, 1×10⁻¹⁵ nm, or less. Thelasers may emit light having a bandwidth that is within a range definedby any two of the preceding values.

The light sources may be configured to emit light tuned to one or moremagic wavelengths corresponding to the plurality of atoms. A magicwavelength corresponding to an atom may comprise any wavelength of lightthat gives rise to equal or nearly equal polarizabilities of the firstand second atomic states. The magic wavelengths for a transition betweenthe first and second atomic states may be determined by calculating thewavelength-dependent polarizabilities of the first and second atomicstates and finding crossing points. Light tuned to such a magicwavelength may give rise to equal or nearly equal differential lightshifts in the first and second atomic states, regardless of theintensity of the light emitted by the light sources. This mayeffectively decouple the first and second atomic states from motion ofthe atoms. The magic wavelengths may utilize one or more scalar ortensor light shifts. The scalar or tensor light shifts may depend onmagnetic sublevels within the first and second atomic states.

For instance, Group II atoms and metastable states of alkaline earth oralkaline earth-like atoms may possess relatively large tensor shiftswhose angle relative to an applied magnetic field may be tuned to causea situation in which scalar and tensor shifts balance and give a zero ornear zero differential light shift between the first and second atomicstates. The angle θ may be tuned by selecting the polarization of theemitted light. For instance, when the emitted light is linearlypolarized, the total polarizability α may be written as a sum of thescalar component α_(scalar) and the tensor component α_(tensor):α=α_(scalar)+(3 cos²θ−1)α_(tensor)

By choosing θ appropriately, the polarizability of the first and secondatomic states may be chosen to be equal or nearly equal, correspondingto a zero or near zero differential light shift, and the motion of theatoms may be decoupled.

The light sources may be configured to direct light to one or moreoptical modulators (OMs) configured to generate the plurality of opticaltrapping sites. For instance, the optical trapping unit may comprise anOM 214 configured to generate the plurality of optical trapping sites.Although depicted as comprising one OM in FIG. 3A, the optical trappingunit may comprise any number of OMs, such as at least about 1, 2, 3, 4,5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3,2, or 1 OMs. The OMs may comprise one or more digital micromirrordevices (DMDs). The OMs may comprise one or more liquid crystal devices,such as one or more liquid crystal on silicon (LCoS) devices. The OMsmay comprise one or more spatial light modulators (SLMs). The OMs maycomprise one or more acousto-optic deflectors (AODs) or acousto-opticmodulators (AOMs). The OMs may comprise one or more electro-opticdeflectors (EODs) or electro-optic modulators (EOMs).

The OM may be optically coupled to one or more optical element togenerate a regular array of optical trapping sites. For instance, the OMmay be optically coupled to optical element 219, as shown in FIG. 3A.The optical elements may comprise lenses or microscope objectivesconfigured to re-direct light from the OMs to form a regular rectangulargrid of optical trapping sites.

For instance, as shown in FIG. 3A, the OM may comprise an SLM, DMD, orLCoS device. The SLM, DMD, or LCoS device may be imaged onto the backfocal plane of the microscope objectives. This may allow for thegeneration of an arbitrary configuration of optical trapping sites intwo or three dimensions.

Alternatively or in combination, the OMs may comprise first and secondAODs. The active regions of the first and second AODs may be imaged ontothe back focal plane of the microscope objectives. The output of thefirst AOD may be optically coupled to the input of the second AOD. Inthis manner, the second AOD may make a copy of the optical output of thefirst AOD. This may allow for the generation of optical trapping sitesin two or three dimensions.

Alternatively or in combination, the OMs may comprise static opticalelements, such as one or more microlens arrays or holographic opticalelements. The static optical elements may be imaged onto the back focalplane of the microscope objectives. This may allow for the generation ofan arbitrary configuration of optical trapping sites in two or threedimensions.

The optical trapping unit may comprise one or more imaging unitsconfigured to obtain one or more images of a spatial configuration ofthe plurality of atoms trapped within the optical trapping sites. Forinstance, the optical trapping unit may comprise imaging unit 215.Although depicted as comprising a single imaging unit in FIG. 3A, theoptical trapping unit may comprise any number of imaging units, such asat least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more imaging units orat most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 imaging units. Theimaging units may comprise one or more lens or objectives. The imagingunits may comprise one or more PMTs, photodiodes, avalanche photodiodes,phototransistors, reverse-biased LEDs, CCDs, or CMOS cameras. Theimaging unit may comprise one or more fluorescence detectors. The imagesmay comprise one or more fluorescence images, single-atom fluorescenceimages, absorption images, single-atom absorption images, phase contrastimages, or single-atom phase contrast images.

The optical trapping unit may comprise one or more spatial configurationartificial intelligence (AI) units configured to perform one or more AIoperations to determine the spatial configuration of the plurality ofatoms trapped within the optical trapping sites based on the imagesobtained by the imaging unit. For instance, the optical trapping unitmay comprise spatial configuration AI unit 216. Although depicted ascomprising a single spatial configuration AI unit in FIG. 3A, theoptical trapping unit may comprise any number of spatial configurationAI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or morespatial configuration AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3,2, or 1 spatial configuration AI units. The AI operations may compriseany machine learning (ML) or reinforcement learning (RL) operationsdescribed herein.

The optical trapping unit may comprise one or more atom rearrangementunits configured to impart an altered spatial arrangement of theplurality of atoms trapped with the optical trapping sites based on theone or more images obtained by the imaging unit. For instance, theoptical trapping unit may comprise atom rearrangement unit 217. Althoughdepicted as comprising a single atom rearrangement unit in FIG. 3A, theoptical trapping unit may comprise any number of atom rearrangementunits, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or moreatom rearrangement units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or1 atom rearrangement units.

The optical trapping unit may comprise one or more spatial arrangementartificial intelligence (AI) units configured to perform one or more AIoperations to determine the altered spatial arrangement of the pluralityof atoms trapped within the optical trapping sites based on the imagesobtained by the imaging unit. For instance, the optical trapping unitmay comprise spatial arrangement AI unit 218. Although depicted ascomprising a single spatial arrangement AI unit in FIG. 3A, the opticaltrapping unit may comprise any number of spatial arrangement AI units,such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatialarrangement AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1spatial arrangement AI units. The AI operations may comprise any machinelearning (ML) or reinforcement learning (RL) operations describedherein.

In some cases, the spatial configuration AI units and the spatialarrangement AI units may be integrated into an integrated AI unit. Theoptical trapping unit may comprise any number of integrated AI units,such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integratedAI units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integratedAI units.

The atom rearrangement unit may be configured to alter the spatialarrangement in order to obtain an increase in a filling factor of theplurality of optical trapping sites. A filling factor may be defined asa ratio of the number of optical trapping sites occupied by one or moreatoms to the total number of optical trapping sites available in theoptical trapping unit or in a region of the optical trapping unit. Forinstance, initial loading of atoms within the optical trapping sites maygive rise to a filling factor of less than 100%, 90%, 80%, 70%, 60%,50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%,50%, or less of the available optical trapping sites, respectively. Itmay be desirable to rearrange the atoms to achieve a filling factor ofat least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing theimaging information obtained by the imaging unit, the atom rearrangementunit may attain a filling factor of at least about 50%, 60%, 70%, 80%,90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%,99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%,99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atomrearrangement unit may attain a filling factor of at most about 99.99%,99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%,99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%,96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atomrearrangement unit may attain a filling factor that is within a rangedefined by any two of the preceding values.

By way of example, FIG. 3C shows an example of an optical trapping unitthat is partially filled with atoms. As depicted in FIG. 3C, initialloading of atoms within the optical trapping sites may give rise to afilling factor of 44.4% (4 atoms filling 9 available optical trappingsites). By moving atoms from different regions of the optical trappingunit (not shown in FIG. 3C) to unoccupied optical trapping sites or bymoving atoms from an atom reservoir described herein, a much higherfilling factor may be obtained, as shown in FIG. 3D.

FIG. 3D shows an example of an optical trapping unit that is completelyfilled with atoms. As depicted in FIG. 3D, fifth atom 212 e, sixth atom212 f, seventh atom 212 g, eighth atom 212 h, and ninth atom 212 i maybe moved to fill unoccupied optical trapping sites. The fifth, sixth,seventh, eighth, and ninth atoms may be moved from different regions ofthe optical trapping unit (not shown in FIG. 3C) or by moving atoms froman atom reservoir described herein. Thus, the filling factor may besubstantially improved following rearrangement of atoms within theoptical trapping sites. For instance, a filling factor of up to 100%(such 9 atoms filling 9 available optical trapping sites, as shown inFIG. 3D) may be attained.

Electromagnetic Delivery Units

FIG. 4 shows an example of an electromagnetic delivery unit 220. Theelectromagnetic delivery unit may be configured to apply electromagneticenergy to one or more atoms of the plurality of atoms, as describedherein. The electromagnetic delivery unit may comprise one or more lightsources, such as any light source described herein. The electromagneticenergy may comprise optical energy. The optical energy may comprise anyrepetition rate, pulse energy, average power, wavelength, or bandwidthdescribed herein.

The electromagnetic delivery unit may comprise one or more microwave orradio-frequency (RF) energy sources, such as one or more magnetrons,klystrons, traveling-wave tubes, gyrotrons, field-effect transistors(FETs), tunnel diodes, Gunn diodes, impact ionization avalanchetransit-time (IMPATT) diodes, or masers. The electromagnetic energy maycomprise microwave energy or RF energy. The RF energy may comprise oneor more wavelengths of at least about 1 millimeter (mm), 2 mm, 3 mm, 4mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm,700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m,9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m,300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km,3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or more. The RF energymay comprise one or more wavelengths of at most about 10 km, 9 km, 8 km,7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m,500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m,30 m, 20 m, 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm,800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm,80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or less. The RF energy maycomprise one or more wavelengths that are within a range defined by anytwo of the preceding values.

The RF energy may comprise an average power of at least about 1microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW,20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW,300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt(mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW,400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 Watt (W), 2 W, 3 W, 4W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800 W, 900 W,1,000 W, or more. The RF energy may comprise an average power of at mostabout 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W,100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW,400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW,1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW,100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. The RFenergy may comprise an average power that is within a range defined byany two of the preceding values.

The electromagnetic delivery unit may comprise one or more lightsources, such as any light source described herein. For instance, theelectromagnetic delivery unit may comprise light source 221. Althoughdepicted as comprising a single light source in FIG. 4 , theelectromagnetic delivery unit may comprise any number of light sources,such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more lightsources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources.

The light sources may be configured to direct light to one or more OMsconfigured to selectively apply the electromagnetic energy to one ormore atoms of the plurality of atoms. For instance, the electromagneticdelivery unit may comprise OM 222. Although depicted as comprising asingle OM in FIG. 4 , the electromagnetic delivery unit may comprise anynumber of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ormore OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMsmay comprise one or more SLMs, AODs, or AOMs. The OMs may comprise oneor more DMDs. The OMs may comprise one or more liquid crystal devices,such as one or more LCoS devices.

The electromagnetic delivery unit may comprise one or moreelectromagnetic energy artificial intelligence (AI) units configured toperform one or more AI operations to selectively apply theelectromagnetic energy to the atoms. For instance, the electromagneticdelivery unit may comprise AI unit 223. Although depicted as comprisinga single AI unit in FIG. 4 , the electromagnetic delivery unit maycomprise any number of AI units, such as at least about 1, 2, 3, 4, 5,6, 7, 8, 9, 10, or more AI units or at most about 10, 9, 8, 7, 6, 5, 4,3, 2, or 1 AI units. The AI operations may comprise any machine learning(ML) or reinforcement learning (RL) operations described herein.

The electromagnetic delivery unit may be configured to apply one or moresingle-qubit operations (such as one or more single-qubit gateoperations) on the qubits described herein. The electromagnetic deliveryunit may be configured to apply one or more two-qubit operations (suchas one or more two-qubit gate operations) on the two-qubit unitsdescribed herein. Each single-qubit or two-qubit operation may comprisea duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns,50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (μs), 2 μs, 3 μs, 4μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60μs, 70 μs, 80 μs, 90 μs, 100 μs, or more. Each single-qubit or two-qubitoperation may comprise a duration of at most about 100 μs, 90 μs, 80 μs,70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs,5 μs, 4 μs, 3 μs, 2 μs, 1 μs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns,400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40ns, 30 ns, 20 ns, 10 ns, or less. Each single-qubit or two-qubitoperation may comprise a duration that is within a range defined by anytwo of the preceding values. The single-qubit or two-qubit operationsmay be applied with a repetition frequency of at least 1 kilohertz(kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz,20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz,200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz,1,000 kHz, or more. The single-qubit or two-qubit operations may beapplied with a repetition frequency of at most 1,000 kHz, 900 kHz, 800kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less.The single-qubit or two-qubit operations may be applied with arepetition frequency that is within a range defined by any two of thepreceding values.

The electromagnetic delivery unit may be configured to apply one or moresingle-qubit operations by inducing one or more Raman transitionsbetween a first qubit state and a second qubit state described herein.The Raman transitions may be detuned from a ³P₀ or ³P₁ line describedherein. For instance, the Raman transitions may be detuned by at leastabout 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900kHz, 1 MHz, 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900MHz, 1 GHz, or more. The Raman transitions may be detuned by at mostabout 1 GHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz,30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz,2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz,30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz,2 kHz, 1 kHz, or less. The Raman transitions may be detuned by a valuethat is within a range defined by any two of the preceding values.

Raman transitions may be induced on individually selected atoms usingone or more spatial light modulators (SLMs) or acousto-optic deflectors(AODs) to impart a deflection angle and/or a frequency shift to a lightbeam based on an applied radio-frequency (RF) signal. The SLM or AOD maybe combined with an optical conditioning system that images the SLM orAOD active region onto the back focal plane of a microscope objective.The microscope objective may perform a spatial Fourier transform on theoptical field at the position of the SLM or AOD. As such, angle (whichmay be proportional to RF frequency) may be converted into position. Forexample, applying a comb of radio frequencies to an AOD may generate alinear array of spots at a focal plane of the objective, with each spothaving a finite extent determined by the characteristics of the opticalconditioning system (such as the point spread function of the opticalconditioning system).

To perform a Raman transition on a single atom with a single SLM or AOD,a pair of frequencies may be applied to the SLM or AOD simultaneously.The two frequencies of the pair may have a frequency difference thatmatches or nearly matches the splitting energy between the first andsecond qubit states. For instance, the frequency difference may differfrom the splitting energy by at most about 1 MHz, 900 kHz, 800 kHz, 700kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz,700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The frequency difference may differfrom the splitting energy by at least about 1 Hz, 2 Hz, 3 Hz, 4 Hz, 5Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700Hz, 800 Hz, 900 Hz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700kHz, 800 kHz, 900 kHz, 1 MHz, or more. The frequency difference maydiffer from the splitting energy by about 0 Hz. The frequency differencemay differ from the splitting energy by a value that is within a rangedefined by any two of the preceding values. The optical system may beconfigured such that the position spacing corresponding to the frequencydifference is not resolved and such that light at both of the twofrequencies interacts with a single atom.

Integrated Optical Trapping Units and Electromagnetic Delivery Units

The optical trapping units and electromagnetic delivery units describedherein may be integrated into a single optical system. A microscopeobjective may be used to deliver electromagnetic radiation generated byan electromagnetic delivery unit described herein and to deliver lightfor trapping atoms generated by an optical trapping unit describedherein. Alternatively or in combination, different objectives may beused to deliver electromagnetic radiation generated by anelectromagnetic delivery unit and to deliver light from trapping atomsgenerated by an optical trapping unit.

A single SLM or AOD may allow the implementation of qubit operations(such as any single-qubit or two-qubit operations described herein) on alinear array of atoms. Alternatively or in combination, two separateSLMs or AODs may be configured to each handle light with orthogonalpolarizations. The light with orthogonal polarizations may be overlappedbefore the microscope objective. In such a scheme, each photon used in atwo-photon transition described herein may be passed to the objective bya separate SLM or AOD, which may allow for increased polarizationcontrol. Qubit operations may be performed on a two-dimensionalarrangement of atoms by bringing light from a first SLM or AOD into asecond SLM or AOD that is oriented substantially orthogonally to thefirst SLM or AOD via an optical relay. Alternatively or in combination,qubit operations may be performed on a two-dimensional arrangement ofatoms by using a one-dimensional array of SLMs or AODs.

The stability of qubit gate fidelity may be improved by maintainingoverlap of light from the various light sources described herein (suchas light sources associated with the optical trapping units orelectromagnetic delivery units described herein). Such overlap may bemaintained by an optical subsystem that measures the direction of lightemitted by the various light sources, allowing closed-loop control ofthe direction of light emission. The optical subsystem may comprise apickoff mirror located before the microscope objective. The pickoffmirror may be configured to direct a small amount of light to a lens,which may focus a collimated beam and convert angular deviation intoposition deviation. A position-sensitive optical detector, such as alateral-effect position sensor or quadrant photodiode, may convert theposition deviation into an electronic signal and information about thedeviation may be fed into a compensation optic, such as an activemirror.

The stability of qubit gate manipulation may be improved by controllingthe intensity of light from the various light sources described herein(such as light sources associated with the optical trapping units orelectromagnetic delivery units described herein). Such intensity controlmay be maintained by an optical subsystem that measures the intensity oflight emitted by the various light sources, allowing closed-loop controlof the intensity. Each light source may be coupled to an intensityactuator, such as an intensity servo control. The actuator may comprisean acousto-optic modulator (AOM) or electro-optic modulator (EOM). Theintensity may be measured using an optical detector, such as aphotodiode or any other optical detector described herein. Informationabout the intensity may be integrated into a feedback loop to stabilizethe intensity.

State Preparation Units

FIG. 5 shows an example of a state preparation unit 250. The statepreparation unit may be configured to prepare a state of the pluralityof atoms, as described herein. The state preparation unit may be coupledto the optical trapping unit and may direct atoms that have beenprepared by the state preparation unit to the optical trapping unit. Thestate preparation unit may be configured to cool the plurality of atoms.The state preparation unit may be configured to cool the plurality ofatoms prior to trapping the plurality of atoms at the plurality ofoptical trapping sites. The atoms may be cooled to a first distributionof atomic states. The first distribution of atomic states may be anon-equilibrium distribution of atomic states.

The state preparation unit may comprise one or more Zeeman slowers. Forinstance, the state preparation unit may comprise a Zeeman slower 251.Although depicted as comprising a single Zeeman slower in FIG. 5 , thestate preparation may comprise any number of Zeeman slowers, such as atleast about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Zeeman slowers or atmost about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Zeeman slowers. The Zeemanslowers may be configured to cool one or more atoms of the plurality ofatoms from a first velocity or distribution of velocities (such anemission velocity from an of an atom source, room temperature, liquidnitrogen temperature, or any other temperature) to a second velocitythat is lower than the first velocity or distribution of velocities.

The first velocity or distribution of velocities may be associated witha temperature of at least about 50 Kelvin (K), 60 K, 70 K, 80 K, 90 K,100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1,000 K,or more. The first velocity or distribution of velocities may beassociated with a temperature of at most about 1,000 K, 900 K, 800 K,700 K, 600 K, 500 K, 400 K, 300 K, 200 K, 100 K, 90 K, 80 K, 70 K, 60 K,50 K, or less. The first velocity or distribution of velocities may beassociated with a temperature that is within a range defined by any twoof the preceding values. The second velocity may be at least about 1meter per second (m/s), 2 m/s, 3 m/s, 4 m/s, 5 m/s, 6 m/s, 7 m/s, 8 m/s,9 m/s, 10 m/s, or more. The second velocity may be at most about 10 m/s,9 m/s, 8 m/s, 7 m/s, 6 m/s, 5 m/s, 4 m/s, 3 m/s, 2 m/s, 1 m/s, or less.The second velocity may be within a range defined by any two of thepreceding values. The Zeeman slowers may comprise 1D Zeeman slowers.

The state preparation unit may comprise a first magneto-optical trap(MOT) 252. The first MOT may be configured to cool the atoms to a firsttemperature. The first temperature may be at most about 10 millikelvin(mK), 9 mK, 8 mK, 7 mK, 6 mK, 5 mK, 4 mK, 3 mK, 2 mK, 1 mK, 0.9 mK, 0.8mK, 0.7 mK, 0.6 mK, 0.5 mK, 0.4 mK, 0.3 mK, 0.2 mK, 0.1 mK, or less. Thefirst temperature may be at least about 0.1 mK, 0.2 mK, 0.3 mK, 0.4 mK,0.5 mK, 0.6 mK, 0.7 mK, 0.8 mK, 0.9 mK, 1 mK, 2 mK, 3 mK, 4 mK, 5 mK, 6mK, 7 mK, 8 mK, 9 mK, 10 mK, or more. The first temperature may bewithin a range defined by any two of the preceding values. The first MOTmay comprise a 1D, 2D, or 3D MOT.

The first MOT may comprise one or more light sources (such as any lightsource described herein) configured to emit light. The light maycomprise one or more wavelengths of at least about 400 nm, 410 nm, 420nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may compriseone or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or morewavelengths that are within a range defined by any two of the precedingvalues. For instance, the light may comprise one or more wavelengthsthat are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm,400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm,or 650 nm to 700 nm.

The state preparation unit may comprise a second MOT 253. The second MOTmay be configured to cool the atoms from the first temperature to asecond temperature that is lower than the first temperature. The secondtemperature may be at most about 100 microkelvin (μK), 90 μK, 80 μK, 70μK, 60 μK, 50 μK, 40 μK, 30 μK, 20 μK, 10 μK, 9 μK, 8 μK, 7 μK, 6 μK, 5μK, 4 μK, 3 μK, 2 μK, 1 μK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK,500 nK, 400 nK, 300 nK, 200 nK, 100 nK, or less. The second temperaturemay be at least about 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK,700 nK, 800 nK, 900 nK, 1 μK, 2 μK, 3 μK, 4 μK, 5 μK, 6 μK, 7 μK, 8 μK,9 μK, 10 μK, 20 μK, 30 μK, 40 μK, 50 μK, 60 μK, 70 μK, 80 μK, 90 μK, 100μK, or more. The second temperature may be within a range defined by anytwo of the preceding values. The second MOT may comprise a 1D, 2D, or 3DMOT.

The second MOT may comprise one or more light sources (such as any lightsource described herein) configured to emit light. The light maycomprise one or more wavelengths of at least about 400 nm, 410 nm, 420nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may compriseone or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or morewavelengths that are within a range defined by any two of the precedingvalues. For instance, the light may comprise one or more wavelengthsthat are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm,400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm,or 650 nm to 700 nm.

Although depicted as comprising two MOTs in FIG. 5 , the statepreparation unit may comprise any number of MOTs, such as at least about1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more MOTs or at most about 10, 9, 8,7, 6, 5, 4, 3, 2, or 1 MOTs.

The state preparation unit may comprise one or more sideband coolingunits or Sisyphus cooling units (such as a sideband cooling unitdescribed in https://arxiv.org/abs/1810.06626 or a Sisyphus cooling unitdescribed in https://arxiv.org/abs/1811.06014, each of which isincorporated herein by reference in its entirety for all purposes). Forinstance, the state preparation unit may comprise sideband cooling unitor Sisyphus cooling unit 254. Although depicted as comprising a singlesideband cooling unit or Sisyphus cooling unit in FIG. 5 , the statepreparation may comprise any number of sideband cooling units orSisyphus cooling units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8,9, 10, or more sideband cooling units or Sisyphus cooling units, or atmost about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband cooling units orSisyphus cooling units. The sideband cooling units or Sisyphus coolingunits may be configured to use sideband cooling to cool the atoms fromthe second temperature to a third temperature that is lower than thesecond temperature. The third temperature may be at most about 10 μK, 9μK, 8 μK, 7 μK, 6 μK, 5 μK, 4 μK, 3 μK, 2 μK, 1 μK, 900 nK, 800 nK, 700nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, 90 nK, 80 nK, 70 nK,60 nK, 50 nK, 40 nK, 30 nK, 20 nK, 10 nK, or less. The third temperaturemay be at most about 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800nK, 900 nK, 1 μK, 2 μK, 3 μK, 4 μK, 5 μK, 6 μK, 7 μK, 8 μK, 9 μK, 10 μK,or more. The third temperature may be within a range defined by any twoof the preceding values.

The sideband cooling units or Sisyphus cooling units may comprise one ormore light sources (such as any light source described herein)configured to emit light. The light may comprise one or more wavelengthsof at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm,1,000 nm, or more. The light may comprise one or more wavelengths of atmost about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, orless. The light may comprise one or more wavelengths that are within arange defined by any two of the preceding values. For instance, thelight may comprise one or more wavelengths that are within a range from400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nmto 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

The state preparation unit may comprise one or more optical pumpingunits. For instance, the state preparation unit may comprise opticalpumping unit 255. Although depicted as comprising a single opticalpumping unit in FIG. 5 , the state preparation may comprise any numberof optical pumping units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8,9, 10, or more optical pumping units, or at most about 10, 9, 8, 7, 6,5, 4, 3, 2, or 1 optical pumping units. The optical pumping units may beconfigured to emit light to optically pump the atoms from the firstdistribution of atomic states to a second distribution of atomic states.The second distribution of atomic states may be a non-equilibriumdistribution of atomic states. For instance, the optical pumping unitsmay be configured to emit light to optically pump the atoms from thefirst distribution of atomic states to a single pure or nearly pureatomic state. A pure or nearly pure atomic state may be characterized ascomprising a significant probability of measuring the atoms to be in thepure or nearly pure atomic state. For instance, there may be aprobability of at least about 90%, 95%, 99%, 99.5%, 99.9%, 99.95%,99.99%, or more of measuring the atoms to be in the pure or nearly pureatomic state. There may be a probability of at most about 99.99%,99.95%, 99.9%, 99.5%, 99%, 95%, 90%, or less of measuring the atoms tobe in the pure or nearly pure atomic state. There may be a probabilityof measuring the atoms to be in the pure or nearly pure atomic statethat is within a range defined by any two of the preceding values. Theoptical pumping units may be configured to emit light to optically pumpthe atoms to a ground atomic state or to any other atomic state. Theoptical pumping units may be configured to optically pump the atomsbetween any two atomic states. The optical pumping units may compriseone or more light sources (such as any light source described herein)configured to emit light. The light may comprise one or more wavelengthsof at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm,1,000 nm, or more. The light may comprise one or more wavelengths of atmost about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, orless. The light may comprise one or more wavelengths that are within arange defined by any two of the preceding values. For instance, thelight may comprise one or more wavelengths that are within a range from400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nmto 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

The state preparation unit may comprise one or more coherent drivingunits. For instance, the state preparation unit may comprise coherentdriving unit 256. Although depicted as comprising a coherent drivingunit in FIG. 5 , the state preparation may comprise any number ofcoherent driving units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8,9, 10, or more coherent driving units or at most about 10, 9, 8, 7, 6,5, 4, 3, 2, or 1 coherent driving units. The coherent driving units maybe configured to coherently drive the atoms from the non-equilibriumatomic state or the non-equilibrium distribution of atomic states to thefirst or second atomic states described herein. Thus, the atoms may beoptically pumped to an atomic state that is convenient to access (forinstance, based on availability of light sources that emit particularwavelengths or based on other factors) and then coherently driven toatomic states described herein that are useful for performing quantumcomputations. The coherent driving units may be configured to induce asingle photon transition between the non-equilibrium state or thenon-equilibrium distribution of atomic states and the first or secondatomic state. The coherent driving units may be configured to induce atwo-photon transition between the non-equilibrium state or thenon-equilibrium distribution of atomic states and the first or secondatomic state. The two-photon transition may be induced using light fromtwo light sources described herein (such as two lasers describedherein).

The coherent driving units may comprise one or more light sources (suchas any light source described herein) configured to emit light. Thelight may comprise one or more wavelengths of at least about 400 nm, 410nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light maycomprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise oneor more wavelengths that are within a range defined by any two of thepreceding values. For instance, the light may comprise one or morewavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500nm to 700 nm, or 650 nm to 700 nm.

The coherent driving units may be configured to induce an RF transitionbetween the non-equilibrium state or the non-equilibrium distribution ofatomic states and the first or second atomic state. The coherent drivingunits may comprise one or more electromagnetic radiation sourcesconfigured to emit electromagnetic radiation configured to induce the RFtransition. For instance, the coherent driving units may comprise one ormore RF sources (such as any RF source described herein) configured toemit RF radiation. The RF radiation may comprise one or more wavelengthsof at least about 10 centimeters (cm), 20 cm, 30 cm, 40 cm, 50 cm, 60cm, 70 cm, 80 cm, 90 cm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m,9 m, 10 m, or more. The RF radiation may comprise one or morewavelengths of at most about 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2m, 1 m, 90 cm, 80 cm, 70 cm, 60 cm, 50 cm, 40 cm, 30 cm, 20 cm, 10 cm,or less. The RF radiation may comprise one or more wavelengths that arewithin a range defined by any two of the preceding values. Alternativelyor in combination, the coherent driving units may comprise one or morelight sources (such as any light sources described herein) configured toinduce a two-photon transition corresponding to the RF transition.

Controllers

The optical trapping units, electromagnetic delivery units, entanglementunits, readout optical units, vacuum units, imaging units, spatialconfiguration AI units, spatial arrangement AI units, atom rearrangementunits, state preparation units, sideband cooling units, optical pumpingunits, coherent driving units, electromagnetic energy AI units, atomreservoirs, atom movement units, or Rydberg excitation units may includeone or more circuits or controllers (such as one or more electroniccircuits or controllers) that is connected (for instance, by one or moreelectronic connections) to the optical trapping units, electromagneticdelivery units, entanglement units, readout optical units, vacuum units,imaging units, spatial configuration AI units, spatial arrangement AIunits, atom rearrangement units, state preparation units, sidebandcooling units, optical pumping units, coherent driving units,electromagnetic energy AI units, atom reservoirs, atom movement units,or Rydberg excitation units. The circuits or controllers may beconfigured to control the optical trapping units, electromagneticdelivery units, entanglement units, readout optical units, vacuum units,imaging units, spatial configuration AI units, spatial arrangement AIunits, atom rearrangement units, state preparation units, sidebandcooling units, optical pumping units, coherent driving units,electromagnetic energy AI units, atom reservoirs, atom movement units,or Rydberg excitation units.

Non-Classical Computers

In an aspect, the present disclosure provides a non-classical computercomprising: a plurality of qubits comprising greater than 60 atoms, eachatom trapped within an optical trapping site of a plurality of spatiallydistinct optical trapping sites, wherein the plurality of qubitscomprise at least a first qubit state and a second qubit state, whereinthe first qubit state comprises a first atomic state and the secondqubit state comprises a second atomic state; one or more electromagneticdelivery units configured to apply electromagnetic energy to one or morequbits of the plurality of qubits, thereby imparting a non-classicaloperation to the one or more qubits, which non-classical operationincludes a superposition between at least the first qubit state and thesecond qubit state; one or more entanglement units configured to quantummechanically entangle at least a subset of the plurality of qubits inthe superposition with at least another qubit of the plurality ofqubits; and one or more readout optical units configured to perform oneor more measurements of the one or more qubits, thereby obtaining anon-classical computation.

In an aspect, the present disclosure provide a non-classical computercomprising a plurality of qubits comprising greater than 60 atoms eachtrapped within an optical trapping site of a plurality of spatiallydistinct optical trapping sites.

Methods for Performing a Non-Classical Computation

In an aspect, the present disclosure provides a method for performing anon-classical computation, comprising: (a) generating a plurality ofspatially distinct optical trapping sites, the plurality of opticaltrapping sites configured to trap a plurality of atoms, the plurality ofatoms comprising greater than 60 atoms; (b) applying electromagneticenergy to one or more atoms of the plurality of atoms, thereby inducingthe one or more atoms to adopt one or more superposition states of afirst atomic state and at least a second atomic state that is differentfrom the first atomic state; (c) quantum mechanically entangling atleast a subset of the one or more atoms in the one or more superpositionstates with at least another atom of the plurality of atoms; and (d)performing one or more optical measurements of the one or moresuperposition state to obtain the non-classical computation.

FIG. 6 shows a flowchart for an example of a first method 600 forperforming a non-classical computation.

In a first operation 610, the method 600 may comprise generating aplurality of spatially distinct optical trapping sites. The plurality ofoptical trapping sites may be configured to trap a plurality of atoms.The plurality of atoms may comprise greater than 60 atoms. The opticaltrapping sites may comprise any optical trapping sites described herein.The atoms may comprise any atoms described herein.

In a second operation 620, the method 600 may comprise applyingelectromagnetic energy to one or more atoms of the plurality of atoms,thereby inducing the one or more atoms to adopt one or moresuperposition states of a first atomic state and at least a secondatomic state that is different from the first atomic state. Theelectromagnetic energy may comprise any electromagnetic energy describedherein. The first atomic state may comprise any first atomic statedescribed herein. The second atomic state may comprise any second atomicstate described herein.

In a third operation 630, the method 600 may comprise quantummechanically entangling at least a subset of the one or more atoms inthe one or more superposition states with at least another atom of theplurality of atoms. The atoms may be quantum mechanically entangled inany manner described herein (for instance, as described herein withrespect to FIG. 2 ).

In a fourth operation 640, the method 600 may comprise performing one ormore optical measurements of the one or more superposition state toobtain the non-classical computation. The optical measurements maycomprise any optical measurements described herein.

In an aspect, the present disclosure provides a method for performing anon-classical computation, comprising: (a) providing a plurality ofqubits comprising greater than 60 atoms, each atom trapped within anoptical trapping site of a plurality of spatially distinct opticaltrapping sites, wherein the plurality of qubits comprise at least afirst qubit state and a second qubit state, wherein the first qubitstate comprises a first atomic state and the second qubit statecomprises a second atomic state; (b) applying electromagnetic energy toone or more qubits of the plurality of qubits, thereby imparting anon-classical operation to the one or more qubits, which non-classicaloperation includes a superposition between at least the first qubitstate and the second qubit state; (c) quantum mechanically entangling atleast a subset of the plurality of qubits in the superposition with atleast another qubit of the plurality of qubits; and (d) performing oneor more optical measurements of the one or more qubits, therebyobtaining the-classical computation.

FIG. 7 shows a flowchart for an example of a second method 700 forperforming a non-classical computation.

In a first operation 710, the method 700 may comprise providing aplurality of qubits comprising greater than 60 atoms, each atom trappedwithin an optical trapping site of a plurality of spatially distinctoptical trapping sites, wherein the plurality of qubits comprise atleast a first qubit state and a second qubit state, wherein the firstqubit state comprises a first atomic state and the second qubit statecomprises a second atomic state. The optical trapping sites may compriseany optical trapping sites described herein. The qubits may comprise anyqubits described herein. The atoms may comprise any atoms describedherein. The first qubit state may comprise any first qubit statedescribed herein. The second qubit state may comprise any second qubitstate described herein. The first atomic state may comprise any firstatomic state described herein. The second atomic state may comprise anysecond atomic state described herein.

In a second operation 720, the method 700 may comprise applyingelectromagnetic energy to one or more qubits of the plurality of qubits,thereby imparting a non-classical operation to the one or more qubits,which non-classical operation includes a superposition between at leastthe first qubit state and the second qubit state. The electromagneticenergy may comprise any electromagnetic energy described herein.

In a third operation 730, the method 700 may comprise quantummechanically entangling at least a subset of the plurality of qubits inthe superposition with at least another qubit of the plurality ofqubits. The qubits may be quantum mechanically entangled in any mannerdescribed herein (for instance, as described herein with respect to FIG.2 ).

In a fourth operation 740, the method 700 may comprise performing one ormore optical measurements of the one or more qubits, thereby obtainingthe non-classical computation. The optical measurements may comprise anyoptical measurements described herein.

In an aspect, the present disclosure provide a method for performing anon-classical computation, comprising: (a) providing a plurality ofqubits comprising greater than 60 atoms each trapped within an opticaltrapping site of a plurality of spatially distinct optical trappingsites, and (b) using at least a subset of the plurality of qubits toperform the non-classical computation.

FIG. 8 shows a flowchart for an example of a third method 800 forperforming a non-classical computation.

In a first operation 810, the method 800 may comprise providing aplurality of qubits comprising greater than 60 atoms each trapped withinan optical trapping site of a plurality of spatially distinct opticaltrapping sites. The qubits may comprise any qubits described herein. Theatoms may comprise any atoms described herein. The optical trappingsites may comprise any optical trapping sites described herein.

In a second operation 820, the method 800 may comprise using at least asubset of the plurality of qubits to perform a non-classicalcomputation.

Computer Systems

FIG. 1 shows a computer system 101 that is programmed or otherwiseconfigured to operate any method or system described herein (such assystem or method for performing a non-classical computation describedherein). The computer system 101 can regulate various aspects of thepresent disclosure. The computer system 101 can be an electronic deviceof a user or a computer system that is remotely located with respect tothe electronic device. The electronic device can be a mobile electronicdevice.

The computer system 101 includes a central processing unit (CPU, also“processor” and “computer processor” herein) 105, which can be a singlecore or multi core processor, or a plurality of processors for parallelprocessing. The computer system 101 also includes memory or memorylocation 110 (e.g., random-access memory, read-only memory, flashmemory), electronic storage unit 115 (e.g., hard disk), communicationinterface 120 (e.g., network adapter) for communicating with one or moreother systems, and peripheral devices 125, such as cache, other memory,data storage and/or electronic display adapters. The memory 110, storageunit 115, interface 120 and peripheral devices 125 are in communicationwith the CPU 105 through a communication bus (solid lines), such as amotherboard. The storage unit 115 can be a data storage unit (or datarepository) for storing data. The computer system 101 can be operativelycoupled to a computer network (“network”) 130 with the aid of thecommunication interface 120. The network 130 can be the Internet, aninternet and/or extranet, or an intranet and/or extranet that is incommunication with the Internet. The network 130 in some cases is atelecommunication and/or data network. The network 130 can include oneor more computer servers, which can enable distributed computing, suchas cloud computing. The network 130, in some cases with the aid of thecomputer system 101, can implement a peer-to-peer network, which mayenable devices coupled to the computer system 101 to behave as a clientor a server.

The CPU 105 can execute a sequence of machine-readable instructions,which can be embodied in a program or software. The instructions may bestored in a memory location, such as the memory 110. The instructionscan be directed to the CPU 105, which can subsequently program orotherwise configure the CPU 105 to implement methods of the presentdisclosure. Examples of operations performed by the CPU 105 can includefetch, decode, execute, and writeback.

The CPU 105 can be part of a circuit, such as an integrated circuit. Oneor more other components of the system 101 can be included in thecircuit. In some cases, the circuit is an application specificintegrated circuit (ASIC).

The storage unit 115 can store files, such as drivers, libraries andsaved programs. The storage unit 115 can store user data, e.g., userpreferences and user programs. The computer system 101 in some cases caninclude one or more additional data storage units that are external tothe computer system 101, such as located on a remote server that is incommunication with the computer system 101 through an intranet or theInternet.

The computer system 101 can communicate with one or more remote computersystems through the network 130. For instance, the computer system 101can communicate with a remote computer system of a user. Examples ofremote computer systems include personal computers (e.g., portable PC),slate or tablet PC's (e.g., Apple® iPad, Samsung® Galaxy Tab),telephones, Smart phones (e.g., Apple® iPhone, Android-enabled device,Blackberry®), or personal digital assistants. The user can access thecomputer system 101 via the network 130.

Methods as described herein can be implemented by way of machine (e.g.,computer processor) executable code stored on an electronic storagelocation of the computer system 101, such as, for example, on the memory110 or electronic storage unit 115. The machine executable or machinereadable code can be provided in the form of software. During use, thecode can be executed by the processor 105. In some cases, the code canbe retrieved from the storage unit 115 and stored on the memory 110 forready access by the processor 105. In some situations, the electronicstorage unit 115 can be precluded, and machine-executable instructionsare stored on memory 110.

The code can be pre-compiled and configured for use with a machinehaving a processor adapted to execute the code, or can be compiledduring runtime. The code can be supplied in a programming language thatcan be selected to enable the code to execute in a pre-compiled oras-compiled fashion.

Aspects of the systems and methods provided herein, such as the computersystem 101, can be embodied in programming. Various aspects of thetechnology may be thought of as “products” or “articles of manufacture”typically in the form of machine (or processor) executable code and/orassociated data that is carried on or embodied in a type of machinereadable medium. Machine-executable code can be stored on an electronicstorage unit, such as memory (e.g., read-only memory, random-accessmemory, flash memory) or a hard disk. “Storage” type media can includeany or all of the tangible memory of the computers, processors or thelike, or associated modules thereof, such as various semiconductormemories, tape drives, disk drives and the like, which may providenon-transitory storage at any time for the software programming. All orportions of the software may at times be communicated through theInternet or various other telecommunication networks. Suchcommunications, for example, may enable loading of the software from onecomputer or processor into another, for example, from a managementserver or host computer into the computer platform of an applicationserver. Thus, another type of media that may bear the software elementsincludes optical, electrical and electromagnetic waves, such as usedacross physical interfaces between local devices, through wired andoptical landline networks and over various air-links. The physicalelements that carry such waves, such as wired or wireless links, opticallinks or the like, also may be considered as media bearing the software.As used herein, unless restricted to non-transitory, tangible “storage”media, terms such as computer or machine “readable medium” refer to anymedium that participates in providing instructions to a processor forexecution.

Hence, a machine readable medium, such as computer-executable code, maytake many forms, including but not limited to, a tangible storagemedium, a carrier wave medium or physical transmission medium.Non-volatile storage media include, for example, optical or magneticdisks, such as any of the storage devices in any computer(s) or thelike, such as may be used to implement the databases, etc. shown in thedrawings. Volatile storage media include dynamic memory, such as mainmemory of such a computer platform. Tangible transmission media includecoaxial cables; copper wire and fiber optics, including the wires thatcomprise a bus within a computer system. Carrier-wave transmission mediamay take the form of electric or electromagnetic signals, or acoustic orlight waves such as those generated during radio frequency (RF) andinfrared (IR) data communications. Common forms of computer-readablemedia therefore include for example: a floppy disk, a flexible disk,hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD orDVD-ROM, any other optical medium, punch cards paper tape, any otherphysical storage medium with patterns of holes, a RAM, a ROM, a PROM andEPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wavetransporting data or instructions, cables or links transporting such acarrier wave, or any other medium from which a computer may readprogramming code and/or data. Many of these forms of computer readablemedia may be involved in carrying one or more sequences of one or moreinstructions to a processor for execution.

The computer system 101 can include or be in communication with anelectronic display 135 that comprises a user interface (UI) 140.Examples of UI's include, without limitation, a graphical user interface(GUI) and web-based user interface.

Methods and systems of the present disclosure can be implemented by wayof one or more algorithms. An algorithm can be implemented by way ofsoftware upon execution by the central processing unit 105. Thealgorithm can, for example, implement methods for performing anon-classical computation described herein.

While preferred embodiments of the present invention have been shown anddescribed herein, it will be obvious to those skilled in the art thatsuch embodiments are provided by way of example only. Numerousvariations, changes, and substitutions will now occur to those skilledin the art without departing from the invention. It should be understoodthat various alternatives to the embodiments of the invention describedherein may be employed in practicing the invention. It is intended thatthe following claims define the scope of the invention and that methodsand structures within the scope of these claims and their equivalents becovered thereby.

What is claimed is:
 1. A method for performing a non-classicalcomputation, comprising: (a) providing a plurality of optical trappingsites comprising a plurality of atoms, which plurality of atoms is aplurality of qubits, wherein said plurality of atoms comprises an atomcomprising two valence electrons; (b) adjusting a splitting between atleast a first atomic state and a second atomic state on a ground statemanifold of said atom of said plurality of atoms using a magnetic field;(c) applying electromagnetic energy to one or more atoms of saidplurality of atoms to perform a sequence of qubit gate operations,wherein a qubit gate operation within said sequence of qubit gateoperations comprises: (i) inducing said one or more atoms to adopt oneor more superposition states of said first atomic state and at leastsaid second atomic state that is different from said first atomic state;or (ii) inducing said one or more atoms to adopt one or moresuperposition states of said first atomic state and at least said secondatomic state that is different from said first atomic state, and quantummechanically entangling an atom of said one or more atoms in said one ormore superposition states with another atom of said plurality of atoms;and (d) performing one or more measurements of said one or moresuperposition states to perform said non-classical computation, whereinsaid non-classical computation is encoded in said sequence of qubit gateoperations.
 2. The method of claim 1, wherein said plurality of opticaltrapping sites comprises a plurality of spatially distinct opticaltrapping sites.
 3. The method of claim 1, further comprising, prior to(b), obtaining one or more images of an initial spatial arrangement ofsaid plurality of atoms.
 4. The method of claim 3, further comprising,prior to (b), performing one or more spatial arrangement artificialintelligence (AI) operations to determine said initial spatialarrangement of said plurality of atoms based on said one or more images.5. The method of claim 4, wherein said one or more spatial arrangementAI operations comprise one or more machine learning (ML) operations orreinforcement learning (RL) operations.
 6. The method of claim 4,further comprising, prior to (b), performing one or more spatialre-arrangement AI operations to determine an altered spatial arrangementof said plurality of atoms based on said one or more images.
 7. Themethod of claim 6, wherein said one or more spatial re-arrangement AIoperations comprise one or more ML operations or RL operations.
 8. Themethod of claim 6, wherein (b) comprises moving said one or more atomsof said plurality of atoms based on said altered spatial arrangement. 9.The method of claim 1, wherein (b) comprises using one or moreelectrically tunable lenses, acousto-optic deflectors (AODs),acousto-optic modulators (AOMs), spatial light modulators (SLMs),electro-optic deflectors (EODs), electro-optic modulators (EOMs),digital micromirror devices (DMDs), liquid crystal devices, or liquidcrystal on silicon (LCoS) devices to move said one or more atoms of saidplurality of atoms.
 10. The method of claim 1, wherein moving said oneor more atoms of said plurality of atoms increases a filling factor ofsaid plurality of optical trapping sites.
 11. The method of claim 10,wherein said filling factor comprises a value of at least 70%.
 12. Themethod of claim 1, wherein said plurality of qubits comprises at least10 qubits.
 13. The method of claim 1, wherein (c) comprises using one ormore optical modulators to apply said electromagnetic energy to said oneor more atoms of said plurality of atoms.
 14. The method of claim 13,wherein said one or more optical modulators comprise one or more membersselected from the group consisting of: spatial light modulators (SLMs),acousto-optic deflectors (AODs), acousto-optic modulators (AOMs),electro-optic deflectors (EODs), electro-optic modulators (EOMs),digital micromirror devices (DMDs), liquid crystal devices, and liquidcrystal on silicon (LCoS) devices.
 15. The method of claim 13, wherein(c) comprises using said one or more optical modulators to selectivelyapply said electromagnetic energy to said one or more atoms of saidplurality of atoms.
 16. The method of claim 15, wherein (c) furthercomprises performing one or more electromagnetic energy AI operations toselectively apply said electromagnetic energy to said one or more atomsof said plurality of atoms.
 17. The method of claim 16, wherein said oneor more electromagnetic energy AI operations comprise one or more MLoperations or RL operations.
 18. The method of claim 1, furthercomprising, prior to (c), selecting an individual atom of said one ormore atoms from said plurality of atoms upon which to perform a gateoperation of said sequence of qubit gate operations.
 19. The method ofclaim 18, wherein said gate operation of said sequence of qubit gateoperations comprises said applying said electromagnetic energy to saidone or more atoms.
 20. The method of claim 1, wherein said atom of saidplurality of atoms is an alkaline earth atom.
 21. The method of claim 1,wherein said atom of said plurality of atoms is strontium or ytterbium.22. The method of claim 1, wherein said atom of said plurality of atomsis a neutral atom.